34:I[651451,["/_next/static/chunks/3j194e876w45f.js?dpl=nogit-1790227106","/_next/static/chunks/060-i_62-vse1.js?dpl=nogit-1790227106","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790227106","/_next/static/chunks/2yhf9vq09dqoa.js?dpl=nogit-1790227106","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790227106","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790227106","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790227106","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790227106","/_next/static/chunks/316m85d5s15pp.js?dpl=nogit-1790227106","/_next/static/chunks/2_itvjgqsumf8.js?dpl=nogit-1790227106","/_next/static/chunks/0be1c60akah5d.js?dpl=nogit-1790227106","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790227106","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790227106","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790227106","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790227106","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790227106","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790227106","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790227106","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790227106","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790227106","/_next/static/chunks/1bbv7o2-m-0ni.js?dpl=nogit-1790227106","/_next/static/chunks/097slu7xaeast.js?dpl=nogit-1790227106","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790227106","/_next/static/chunks/117yqm55h4y_p.js?dpl=nogit-1790227106","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790227106","/_next/static/chunks/2eoustw-nfu98.js?dpl=nogit-1790227106","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790227106","/_next/static/chunks/11ryphth5_v5b.js?dpl=nogit-1790227106","/_next/static/chunks/0gvj8_2lj8s8m.js?dpl=nogit-1790227106","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790227106","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790227106","/_next/static/chunks/2ydqdr10ndcei.js?dpl=nogit-1790227106","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790227106","/_next/static/chunks/1br6ag7ig1vnn.js?dpl=nogit-1790227106","/_next/static/chunks/1bnld2zwzjlzx.js?dpl=nogit-1790227106","/_next/static/chunks/09_ag440ieydu.js?dpl=nogit-1790227106","/_next/static/chunks/3_o3al2nkyqic.js?dpl=nogit-1790227106","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790227106","/_next/static/chunks/2m_zh5-tarne0.js?dpl=nogit-1790227106","/_next/static/chunks/3kvhppczyvgp2.js?dpl=nogit-1790227106","/_next/static/chunks/2fk-2o4srnsgz.js?dpl=nogit-1790227106","/_next/static/chunks/00ol_16j3a2oa.js?dpl=nogit-1790227106","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790227106","/_next/static/chunks/2srfni4gkx9g6.js?dpl=nogit-1790227106","/_next/static/chunks/21sij0j_ujna2.js?dpl=nogit-1790227106","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790227106","/_next/static/chunks/2hccrcvndngge.js?dpl=nogit-1790227106","/_next/static/chunks/0ix6b88ybjhi_.js?dpl=nogit-1790227106","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790227106","/_next/static/chunks/1c50trxtlnlob.js?dpl=nogit-1790227106","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790227106","/_next/static/chunks/1jgxupn92tn-_.js?dpl=nogit-1790227106","/_next/static/chunks/2v0ixm9u4yz3r.js?dpl=nogit-1790227106","/_next/static/chunks/3ghbpnibq2fva.js?dpl=nogit-1790227106","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790227106","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790227106","/_next/static/chunks/1y5xmyjgu8grb.js?dpl=nogit-1790227106","/_next/static/chunks/1nnw6yom6-qhh.js?dpl=nogit-1790227106","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790227106","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790227106","/_next/static/chunks/12_xkdi18mnhe.js?dpl=nogit-1790227106","/_next/static/chunks/2wp4esku6l29e.js?dpl=nogit-1790227106","/_next/static/chunks/044q84i54lw83.js?dpl=nogit-1790227106","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790227106","/_next/static/chunks/2e0e0lyynppq_.js?dpl=nogit-1790227106","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790227106","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790227106","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790227106","/_next/static/chunks/2qrlyi_b3nfc0.js?dpl=nogit-1790227106","/_next/static/chunks/2s1hkstzlmj2o.js?dpl=nogit-1790227106"],"UserProgressProvider"] 35:I[196793,["/_next/static/chunks/3j194e876w45f.js?dpl=nogit-1790227106","/_next/static/chunks/060-i_62-vse1.js?dpl=nogit-1790227106","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790227106","/_next/static/chunks/2yhf9vq09dqoa.js?dpl=nogit-1790227106","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790227106","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790227106","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790227106","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790227106","/_next/static/chunks/316m85d5s15pp.js?dpl=nogit-1790227106","/_next/static/chunks/2_itvjgqsumf8.js?dpl=nogit-1790227106","/_next/static/chunks/0be1c60akah5d.js?dpl=nogit-1790227106","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790227106","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790227106","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790227106","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790227106","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790227106","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790227106","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790227106","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790227106","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790227106","/_next/static/chunks/1bbv7o2-m-0ni.js?dpl=nogit-1790227106","/_next/static/chunks/097slu7xaeast.js?dpl=nogit-1790227106","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790227106","/_next/static/chunks/117yqm55h4y_p.js?dpl=nogit-1790227106","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790227106","/_next/static/chunks/2eoustw-nfu98.js?dpl=nogit-1790227106","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790227106","/_next/static/chunks/11ryphth5_v5b.js?dpl=nogit-1790227106","/_next/static/chunks/0gvj8_2lj8s8m.js?dpl=nogit-1790227106","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790227106","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790227106","/_next/static/chunks/2ydqdr10ndcei.js?dpl=nogit-1790227106","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790227106","/_next/static/chunks/1br6ag7ig1vnn.js?dpl=nogit-1790227106","/_next/static/chunks/1bnld2zwzjlzx.js?dpl=nogit-1790227106","/_next/static/chunks/09_ag440ieydu.js?dpl=nogit-1790227106","/_next/static/chunks/3_o3al2nkyqic.js?dpl=nogit-1790227106","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790227106","/_next/static/chunks/2m_zh5-tarne0.js?dpl=nogit-1790227106","/_next/static/chunks/3kvhppczyvgp2.js?dpl=nogit-1790227106","/_next/static/chunks/2fk-2o4srnsgz.js?dpl=nogit-1790227106","/_next/static/chunks/00ol_16j3a2oa.js?dpl=nogit-1790227106","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790227106","/_next/static/chunks/2srfni4gkx9g6.js?dpl=nogit-1790227106","/_next/static/chunks/21sij0j_ujna2.js?dpl=nogit-1790227106","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790227106","/_next/static/chunks/2hccrcvndngge.js?dpl=nogit-1790227106","/_next/static/chunks/0ix6b88ybjhi_.js?dpl=nogit-1790227106","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790227106","/_next/static/chunks/1c50trxtlnlob.js?dpl=nogit-1790227106","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790227106","/_next/static/chunks/1jgxupn92tn-_.js?dpl=nogit-1790227106","/_next/static/chunks/2v0ixm9u4yz3r.js?dpl=nogit-1790227106","/_next/static/chunks/3ghbpnibq2fva.js?dpl=nogit-1790227106","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790227106","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790227106","/_next/static/chunks/1y5xmyjgu8grb.js?dpl=nogit-1790227106","/_next/static/chunks/1nnw6yom6-qhh.js?dpl=nogit-1790227106","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790227106","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790227106","/_next/static/chunks/12_xkdi18mnhe.js?dpl=nogit-1790227106","/_next/static/chunks/2wp4esku6l29e.js?dpl=nogit-1790227106","/_next/static/chunks/044q84i54lw83.js?dpl=nogit-1790227106","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790227106","/_next/static/chunks/2e0e0lyynppq_.js?dpl=nogit-1790227106","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790227106","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790227106","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790227106","/_next/static/chunks/2qrlyi_b3nfc0.js?dpl=nogit-1790227106","/_next/static/chunks/2s1hkstzlmj2o.js?dpl=nogit-1790227106"],"Gingerbread"] 37:I[716297,["/_next/static/chunks/3j194e876w45f.js?dpl=nogit-1790227106","/_next/static/chunks/060-i_62-vse1.js?dpl=nogit-1790227106","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790227106","/_next/static/chunks/2yhf9vq09dqoa.js?dpl=nogit-1790227106","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790227106","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790227106","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790227106","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790227106","/_next/static/chunks/316m85d5s15pp.js?dpl=nogit-1790227106","/_next/static/chunks/2_itvjgqsumf8.js?dpl=nogit-1790227106","/_next/static/chunks/0be1c60akah5d.js?dpl=nogit-1790227106","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790227106","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790227106","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790227106","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790227106","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790227106","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790227106","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790227106","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790227106","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790227106","/_next/static/chunks/1bbv7o2-m-0ni.js?dpl=nogit-1790227106","/_next/static/chunks/097slu7xaeast.js?dpl=nogit-1790227106","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790227106","/_next/static/chunks/117yqm55h4y_p.js?dpl=nogit-1790227106","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790227106","/_next/static/chunks/2eoustw-nfu98.js?dpl=nogit-1790227106","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790227106","/_next/static/chunks/11ryphth5_v5b.js?dpl=nogit-1790227106","/_next/static/chunks/0gvj8_2lj8s8m.js?dpl=nogit-1790227106","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790227106","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790227106","/_next/static/chunks/2ydqdr10ndcei.js?dpl=nogit-1790227106","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790227106","/_next/static/chunks/1br6ag7ig1vnn.js?dpl=nogit-1790227106","/_next/static/chunks/1bnld2zwzjlzx.js?dpl=nogit-1790227106","/_next/static/chunks/09_ag440ieydu.js?dpl=nogit-1790227106","/_next/static/chunks/3_o3al2nkyqic.js?dpl=nogit-1790227106","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790227106","/_next/static/chunks/2m_zh5-tarne0.js?dpl=nogit-1790227106","/_next/static/chunks/3kvhppczyvgp2.js?dpl=nogit-1790227106","/_next/static/chunks/2fk-2o4srnsgz.js?dpl=nogit-1790227106","/_next/static/chunks/00ol_16j3a2oa.js?dpl=nogit-1790227106","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790227106","/_next/static/chunks/2srfni4gkx9g6.js?dpl=nogit-1790227106","/_next/static/chunks/21sij0j_ujna2.js?dpl=nogit-1790227106","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790227106","/_next/static/chunks/2hccrcvndngge.js?dpl=nogit-1790227106","/_next/static/chunks/0ix6b88ybjhi_.js?dpl=nogit-1790227106","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790227106","/_next/static/chunks/1c50trxtlnlob.js?dpl=nogit-1790227106","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790227106","/_next/static/chunks/1jgxupn92tn-_.js?dpl=nogit-1790227106","/_next/static/chunks/2v0ixm9u4yz3r.js?dpl=nogit-1790227106","/_next/static/chunks/3ghbpnibq2fva.js?dpl=nogit-1790227106","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790227106","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790227106","/_next/static/chunks/1y5xmyjgu8grb.js?dpl=nogit-1790227106","/_next/static/chunks/1nnw6yom6-qhh.js?dpl=nogit-1790227106","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790227106","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790227106","/_next/static/chunks/12_xkdi18mnhe.js?dpl=nogit-1790227106","/_next/static/chunks/2wp4esku6l29e.js?dpl=nogit-1790227106","/_next/static/chunks/044q84i54lw83.js?dpl=nogit-1790227106","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790227106","/_next/static/chunks/2e0e0lyynppq_.js?dpl=nogit-1790227106","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790227106","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790227106","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790227106","/_next/static/chunks/2qrlyi_b3nfc0.js?dpl=nogit-1790227106","/_next/static/chunks/2s1hkstzlmj2o.js?dpl=nogit-1790227106"],"TopicRecentlyViewed"] 38:I[647935,["/_next/static/chunks/3j194e876w45f.js?dpl=nogit-1790227106","/_next/static/chunks/060-i_62-vse1.js?dpl=nogit-1790227106","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790227106","/_next/static/chunks/2yhf9vq09dqoa.js?dpl=nogit-1790227106","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790227106","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790227106","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790227106","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790227106","/_next/static/chunks/316m85d5s15pp.js?dpl=nogit-1790227106","/_next/static/chunks/2_itvjgqsumf8.js?dpl=nogit-1790227106","/_next/static/chunks/0be1c60akah5d.js?dpl=nogit-1790227106","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790227106","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790227106","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790227106","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790227106","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790227106","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790227106","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790227106","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790227106","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790227106","/_next/static/chunks/1bbv7o2-m-0ni.js?dpl=nogit-1790227106","/_next/static/chunks/097slu7xaeast.js?dpl=nogit-1790227106","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790227106","/_next/static/chunks/117yqm55h4y_p.js?dpl=nogit-1790227106","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790227106","/_next/static/chunks/2eoustw-nfu98.js?dpl=nogit-1790227106","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790227106","/_next/static/chunks/11ryphth5_v5b.js?dpl=nogit-1790227106","/_next/static/chunks/0gvj8_2lj8s8m.js?dpl=nogit-1790227106","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790227106","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790227106","/_next/static/chunks/2ydqdr10ndcei.js?dpl=nogit-1790227106","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790227106","/_next/static/chunks/1br6ag7ig1vnn.js?dpl=nogit-1790227106","/_next/static/chunks/1bnld2zwzjlzx.js?dpl=nogit-1790227106","/_next/static/chunks/09_ag440ieydu.js?dpl=nogit-1790227106","/_next/static/chunks/3_o3al2nkyqic.js?dpl=nogit-1790227106","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790227106","/_next/static/chunks/2m_zh5-tarne0.js?dpl=nogit-1790227106","/_next/static/chunks/3kvhppczyvgp2.js?dpl=nogit-1790227106","/_next/static/chunks/2fk-2o4srnsgz.js?dpl=nogit-1790227106","/_next/static/chunks/00ol_16j3a2oa.js?dpl=nogit-1790227106","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790227106","/_next/static/chunks/2srfni4gkx9g6.js?dpl=nogit-1790227106","/_next/static/chunks/21sij0j_ujna2.js?dpl=nogit-1790227106","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790227106","/_next/static/chunks/2hccrcvndngge.js?dpl=nogit-1790227106","/_next/static/chunks/0ix6b88ybjhi_.js?dpl=nogit-1790227106","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790227106","/_next/static/chunks/1c50trxtlnlob.js?dpl=nogit-1790227106","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790227106","/_next/static/chunks/1jgxupn92tn-_.js?dpl=nogit-1790227106","/_next/static/chunks/2v0ixm9u4yz3r.js?dpl=nogit-1790227106","/_next/static/chunks/3ghbpnibq2fva.js?dpl=nogit-1790227106","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790227106","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790227106","/_next/static/chunks/1y5xmyjgu8grb.js?dpl=nogit-1790227106","/_next/static/chunks/1nnw6yom6-qhh.js?dpl=nogit-1790227106","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790227106","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790227106","/_next/static/chunks/12_xkdi18mnhe.js?dpl=nogit-1790227106","/_next/static/chunks/2wp4esku6l29e.js?dpl=nogit-1790227106","/_next/static/chunks/044q84i54lw83.js?dpl=nogit-1790227106","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790227106","/_next/static/chunks/2e0e0lyynppq_.js?dpl=nogit-1790227106","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790227106","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790227106","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790227106","/_next/static/chunks/2qrlyi_b3nfc0.js?dpl=nogit-1790227106","/_next/static/chunks/2s1hkstzlmj2o.js?dpl=nogit-1790227106"],"RecordResourceResume"] 39:I[426249,["/_next/static/chunks/3j194e876w45f.js?dpl=nogit-1790227106","/_next/static/chunks/060-i_62-vse1.js?dpl=nogit-1790227106","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790227106","/_next/static/chunks/2yhf9vq09dqoa.js?dpl=nogit-1790227106","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790227106","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790227106","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790227106","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790227106","/_next/static/chunks/316m85d5s15pp.js?dpl=nogit-1790227106","/_next/static/chunks/2_itvjgqsumf8.js?dpl=nogit-1790227106","/_next/static/chunks/0be1c60akah5d.js?dpl=nogit-1790227106","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790227106","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790227106","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790227106","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790227106","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790227106","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790227106","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790227106","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790227106","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790227106","/_next/static/chunks/1bbv7o2-m-0ni.js?dpl=nogit-1790227106","/_next/static/chunks/097slu7xaeast.js?dpl=nogit-1790227106","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790227106","/_next/static/chunks/117yqm55h4y_p.js?dpl=nogit-1790227106","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790227106","/_next/static/chunks/2eoustw-nfu98.js?dpl=nogit-1790227106","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790227106","/_next/static/chunks/11ryphth5_v5b.js?dpl=nogit-1790227106","/_next/static/chunks/0gvj8_2lj8s8m.js?dpl=nogit-1790227106","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790227106","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790227106","/_next/static/chunks/2ydqdr10ndcei.js?dpl=nogit-1790227106","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790227106","/_next/static/chunks/1br6ag7ig1vnn.js?dpl=nogit-1790227106","/_next/static/chunks/1bnld2zwzjlzx.js?dpl=nogit-1790227106","/_next/static/chunks/09_ag440ieydu.js?dpl=nogit-1790227106","/_next/static/chunks/3_o3al2nkyqic.js?dpl=nogit-1790227106","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790227106","/_next/static/chunks/2m_zh5-tarne0.js?dpl=nogit-1790227106","/_next/static/chunks/3kvhppczyvgp2.js?dpl=nogit-1790227106","/_next/static/chunks/2fk-2o4srnsgz.js?dpl=nogit-1790227106","/_next/static/chunks/00ol_16j3a2oa.js?dpl=nogit-1790227106","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790227106","/_next/static/chunks/2srfni4gkx9g6.js?dpl=nogit-1790227106","/_next/static/chunks/21sij0j_ujna2.js?dpl=nogit-1790227106","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790227106","/_next/static/chunks/2hccrcvndngge.js?dpl=nogit-1790227106","/_next/static/chunks/0ix6b88ybjhi_.js?dpl=nogit-1790227106","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790227106","/_next/static/chunks/1c50trxtlnlob.js?dpl=nogit-1790227106","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790227106","/_next/static/chunks/1jgxupn92tn-_.js?dpl=nogit-1790227106","/_next/static/chunks/2v0ixm9u4yz3r.js?dpl=nogit-1790227106","/_next/static/chunks/3ghbpnibq2fva.js?dpl=nogit-1790227106","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790227106","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790227106","/_next/static/chunks/1y5xmyjgu8grb.js?dpl=nogit-1790227106","/_next/static/chunks/1nnw6yom6-qhh.js?dpl=nogit-1790227106","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790227106","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790227106","/_next/static/chunks/12_xkdi18mnhe.js?dpl=nogit-1790227106","/_next/static/chunks/2wp4esku6l29e.js?dpl=nogit-1790227106","/_next/static/chunks/044q84i54lw83.js?dpl=nogit-1790227106","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790227106","/_next/static/chunks/2e0e0lyynppq_.js?dpl=nogit-1790227106","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790227106","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790227106","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790227106","/_next/static/chunks/2qrlyi_b3nfc0.js?dpl=nogit-1790227106","/_next/static/chunks/2s1hkstzlmj2o.js?dpl=nogit-1790227106"],"TopicQuestionBank"] 24:["$","$L34",null,{"bundle":{"topicIds":["7e0865d9-37e1-4f64-a1e9-cc8d221ebcdb","ca86d757-84b8-44c8-9510-43a8290aa2f1","31c5cd90-9577-40f3-8861-c07a9b3c7b0c","54425cac-51d6-415f-a538-deffe0536f9d","b0401270-e130-49d1-857a-8c2956df37f2","d498fb7e-4189-4089-8881-535baad3ce4f","4811743c-d490-487e-bfcf-65451e2798c9","5c3e96b6-a65e-4880-8be7-2b810c59e8ef"],"paperIds":[],"stageIds":[],"stageTopicIds":[],"stageAssessmentIds":[],"topicNoteIds":[]},"children":[["$","$L35",null,{"feature":"question-bank"}],"$L36",["$","$L37",null,{"subjectCode":"a_level_maths__wjec","subjectName":"Maths","topicSlug":"sequences-and-series","topicTitle":"3.4 Sequences and Series (A-level only)","sectionTitle":null,"activity":"worksheet","resourceTitle":"3.4 Sequences and Series (A-level only) Questions","resourceHref":"/a-level/maths/wjec/sequences-and-series/questions/q1","saveTrail":false}],["$","$L38",null,{"href":"/a-level/maths/wjec/sequences-and-series/questions/q1","label":"3.4 Sequences and Series (A-level only)","subjectCode":"a_level_maths__wjec","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 184 – 3.4 Sequences and Series (A-level only) - Questions\",\"description\":\"Practise 3.4 Sequences and Series (A-level only) A Level Maths exam-style questions with worked solutions on MathsGenie.\",\"learningResourceType\":\"Practice questions\",\"educationalLevel\":\"A Level\",\"about\":{\"@type\":\"Thing\",\"name\":\"3.4 Sequences and Series (A-level only)\"},\"url\":\"https://mathsgenie.co.uk/a-level/maths/wjec/sequences-and-series/questions/q184\",\"inLanguage\":\"en-GB\",\"provider\":{\"@id\":\"https://mathsgenie.co.uk/#organization\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L39",null,{"backHref":"/a-level/maths/wjec/questions","basePath":"/a-level/maths/wjec/sequences-and-series/questions","formulaUrl":"https://www.wjec.co.uk/media/4jfdytcz/ss-w27-18-e.pdf","initialIndex":183,"initialQuestionWindow":[{"id":"98d41dcb-08fe-42f2-aa9f-92f939ea317e","revisionId":"3db702e1-48e7-4a28-b785-d1fe747a4f9d","questionNumber":null,"questionType":"LA","specification":"A subscription-based fitness app tracks the number of new premium users each month. In the first three months after a major update, the number of new premium users, in thousands, was recorded as:\n\n\\[ 3k - 28, \\quad k, \\quad k + 6 \\]\n\nwhere \$k\$ is a constant. Given that the number of new users joining each month is modeled as a geometric series,","difficulty":5,"difficultyBand":"medium","tier":null,"totalMarks":9,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"7de09d2e-751b-428a-90d7-0caec45f59dd","label":"a","content":"show that \$k^2 - 5k - 84 = 0\$.","markscheme":"- Since the terms are in a geometric progression, the common ratio $r$ is constant: $\\dfrac{k}{3k - 28} = \\dfrac{k + 6}{k}$ M1\n- Cross-multiply to obtain $k^2 = (3k - 28)(k + 6) = 3k^2 + 18k - 28k - 168$ M1\n- Rearrange to $2k^2 - 10k - 168 = 0$ and divide by 2 to reach $k^2 - 5k - 84 = 0$ A1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"common_ratio","label":"Equating Common Ratios","steps":[{"type":"text","order":0,"title":"Equate the common ratios","content":"For three consecutive terms in a geometric sequence, the ratio between successive terms is constant. This is called the common ratio, $r$.\n\nWe can find two expressions for $r$ using our three terms, $3k - 28$, $k$, and $k + 6$:\n\n1. Divide the second term by the first term:\n$$r = \\frac{k}{3k - 28}$$\n\n2. Divide the third term by the second term:\n$$r = \\frac{k + 6}{k}$$\n\nSince the common ratio is the same, we can equate these two expressions:\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$"},{"type":"text","order":1,"title":"Cross-multiply and expand","content":"To eliminate the fractions, we cross-multiply both sides of the equation:\n\n$$k \\times k = (3k - 28)(k + 6)$$\n\n$$k^2 = (3k - 28)(k + 6)$$\n\nNow, expand the right-hand side using the FOIL method:\n\n$$k^2 = 3k^2 + 18k - 28k - 168$$\n\nSimplify the terms on the right-hand side:\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":2,"title":"Rearrange and simplify","content":"Rearrange the equation to set it equal to zero by subtracting $k^2$ from both sides:\n\n$$0 = 2k^2 - 10k - 168$$\n\nTo match the target equation, we can divide every term in the equation by $2$:\n\n$$0 = k^2 - 5k - 84$$\n\nRearranging gives us the desired quadratic equation:\n\n$$k^2 - 5k - 84 = 0$$"}]},{"id":"geometric_mean","label":"Using Geometric Mean Relation","steps":[{"type":"text","order":0,"title":"Identify terms of the sequence","content":"We are given that the first three terms of a geometric series are:\n\n* $u_1 = 3k - 28$\n* $u_2 = k$\n* $u_3 = k + 6$\n\nwhere $k$ is a constant."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_1 = \\color{@sky}{3k - 28}, \\quad u_2 = \\color{@purple}{k}, \\quad u_3 = \\color{@amber}{k + 6}$"},{"latex":"$$(\\htmlData{anchor=u2-lhs}{\\color{@purple}{u_2}})^2 = \\htmlData{anchor=u1-rhs}{\\color{@sky}{u_1}} \\cdot \\htmlData{anchor=u3-rhs}{\\color{@amber}{u_3}}$"},{"latex":"$$(\\htmlData{anchor=k-lhs}{\\color{@purple}{k}})^2 = (\\htmlData{anchor=u1-sub}{\\color{@sky}{3k - 28}})(\\htmlData{anchor=u3-sub}{\\color{@amber}{k + 6}})$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"u2-lhs"},{"color":"sky","style":"text-color","anchor":"u1-rhs"},{"color":"amber","style":"text-color","anchor":"u3-rhs"}],"connections":[{"to":"k-lhs","from":"u2-lhs","color":"purple","label":"substitute","style":"solid"},{"to":"u1-sub","from":"u1-rhs","color":"sky","label":null,"style":"solid"},{"to":"u3-sub","from":"u3-rhs","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Apply geometric mean relation","description":"In any geometric progression, the ratio between consecutive terms is constant, which means the square of the middle term is equal to the product of the two outer terms: $u_2^2 = u_1 u_3$. This is called the geometric mean relation."},{"type":"text","order":2,"title":"Expand and simplify the equation","content":"Now, let's expand the right-hand side of the equation:\n\n$$(3k - 28)(k + 6) = 3k^2 + 18k - 28k - 168$$\n\nCombine the like terms in the expansion:\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":3,"title":"Rearrange to the final form","content":"To get our quadratic equation into standard form, we move all terms to one side:\n\n$$3k^2 - k^2 - 10k - 168 = 0$$\n\n$$2k^2 - 10k - 168 = 0$$\n\nNow, divide every term by $2$ to simplify to the final requested equation:\n\n$$k^2 - 5k - 84 = 0$$\n\nThis completes the proof as required."}]}],"generatedAt":"2026-06-09T08:04:47.713Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"763732e7-bab0-4f62-a4cf-28baaedd9d5e","label":"b","content":"Predict the number of new premium users joining in the 6th month.","markscheme":"- Solve $k^2 - 5k - 84 = 0 \\Rightarrow (k - 12)(k + 7) = 0$. Since user counts must be positive, $k = 12$ M1\n- Find $a = 3(12) - 28 = 8$ and $r = \\frac{12}{8} = 1.5$. Then $u_6 = 8 \\times (1.5)^{5} = 60.75$. The number of users is $60,750$ A1","marks":2,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"algebraic_formula","label":"Algebraic Equation and Formula","steps":[{"type":"text","order":0,"title":"Set up the ratio equation","content":"For a sequence to be geometric, the ratio between consecutive terms must be constant. This constant is the common ratio, $r$.\n\nUsing the first three terms of our sequence, we can write:\n\n$$r = \\frac{u_2}{u_1} = \\frac{u_3}{u_2}$$\n\nSubstituting the terms given in the question, $u_1 = 3k - 28$, $u_2 = k$, and $u_3 = k + 6$, we get:\n\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$"},{"type":"text","order":1,"title":"Form a quadratic equation","content":"To solve this equation, cross-multiply to eliminate the fractions:\n\n$$k^2 = (k + 6)(3k - 28)$$\n\nExpand the right-hand side of the equation:\n\n$$k^2 = 3k^2 - 28k + 18k - 168$$\n\n$$k^2 = 3k^2 - 10k - 168$$\n\nRearrange everything to one side to form a standard quadratic equation:\n\n$$2k^2 - 10k - 168 = 0$$\n\nDividing the entire equation by $2$ simplifies it to:\n\n$$k^2 - 5k - 84 = 0$$"},{"type":"text","order":2,"title":"Solve for $k$","content":"Now we factorise the quadratic equation to find the values of $k$. We are looking for two numbers that multiply to $-84$ and add up to $-5$. These numbers are $-12$ and $+7$:\n\n$$(k - 12)(k + 7) = 0$$\n\nThis gives two possible solutions for $k$:\n\n$$k = 12 \\quad \\text{or} \\quad k = -7$$\n\nSince the number of new premium users must be positive, the term $u_2 = k$ must be positive. Therefore, we discard $k = -7$ and select:\n\n$$k = 12$$"},{"type":"text","order":3,"title":"Find the first term and ratio","content":"Now substitute $k = 12$ back to find the first term, $a$ (or $u_1$), and the common ratio, $r$:\n\n* First term: \n $$a = 3(12) - 28 = 36 - 28 = 8$$\n* Common ratio: \n $$r = \\frac{k}{a} = \\frac{12}{8} = 1.5$$"},{"type":"text","order":4,"title":"Calculate the 6th term","content":"The general formula for the $n$-th term of a geometric sequence is:\n\n$$u_n = a r^{n-1}$$\n\nTo find the number of users in the 6th month ($u_6$):\n\n$$u_6 = 8 \\times (1.5)^{6-1}$$\n\n$$u_6 = 8 \\times (1.5)^5$$\n\n$$u_6 = 8 \\times 7.59375 = 60.75$$"},{"type":"text","order":5,"title":"Convert to the final unit","content":"The question specifies that the number of premium users is tracked **in thousands**. \n\nTherefore, we multiply our result by $1,000$ to get the final prediction:\n\n$$\\text{New premium users} = 60.75 \\times 1,000 = 60,750$$"}]},{"id":"recursive_calculation","label":"Step-by-Step Multiplication","steps":[{"type":"text","order":0,"title":"Set up the equation for $k$","content":"Since the number of new users joining each month forms a geometric sequence, the ratio between consecutive terms must be constant.\n\nThis means the common ratio $r$ can be written as:\n\n$$r = \\frac{\\text{Term 2}}{\\text{Term 1}} = \\frac{\\text{Term 3}}{\\text{Term 2}}$$\n\nSubstituting our terms $3k - 28$, $k$, and $k + 6$:\n\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$\n\nNow, we cross-multiply to solve for $k$:\n\n$$k^2 = (k + 6)(3k - 28)$$\n\nExpanding the right-hand side:\n\n$$k^2 = 3k^2 - 28k + 18k - 168$$\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":1,"title":"Solve the quadratic equation","content":"Rearranging the equation to standard quadratic form $ax^2 + bx + c = 0$ by subtracting $k^2$ from both sides:\n\n$$2k^2 - 10k - 168 = 0$$\n\nDividing the entire equation by $2$ to simplify:\n\n$$k^2 - 5k - 84 = 0$$\n\nNow we factorise this quadratic. We need two numbers that multiply to $-84$ and add to $-5$. These numbers are $-12$ and $+7$:\n\n$$(k - 12)(k + 7) = 0$$\n\nThis gives two possible values for $k$:\n\n$$k = 12 \\quad \\text{or} \\quad k = -7$$\n\nSince the number of users joining must be positive (for example, Term 2 is $k$), we must choose the positive value:\n\n$$k = 12$$","calculator":[{"gdc":{"instructions":"Go to the Equation menu, select Polynomial (F2), choose Degree 2, enter coefficients 1, -5, and -84, then press SOLVE (F6)."},"ti84":{"instructions":"Press [APPS], select 'PlySmlt2' (Polynomial Root Finder and Simultaneous Equation Solver). Select 'Poly Root Finder', set order to 2, and enter coefficients: a = 1, b = -5, c = -84. Press [SOLVE]."},"desmos":{"expression":"y = x^2 - 5x - 84","instructions":"Graph the equation y = x^2 - 5x - 84 and identify the x-intercepts."},"description":"Verify the roots of the quadratic equation $k^2 - 5k - 84 = 0$ using a solver."}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$k = \\htmlData{anchor=k-val}{\\color{@sky}{12}}$"},{"latex":"$$\\text{Term 1: } 3(\\htmlData{anchor=k-sub1}{\\color{@sky}{12}}) - 28 = \\htmlData{anchor=t1}{\\color{@amber}{8}}$"},{"latex":"$$\\text{Term 2: } \\htmlData{anchor=t2}{\\color{@purple}{12}}$"},{"latex":"$$\\text{Term 3: } \\htmlData{anchor=k-sub3}{\\color{@sky}{12}} + 6 = \\htmlData{anchor=t3}{\\color{@green}{18}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"k-val"},{"color":"amber","style":"box","anchor":"t1"},{"color":"purple","style":"box","anchor":"t2"},{"color":"green","style":"box","anchor":"t3"}],"connections":[{"to":"k-sub1","from":"k-val","color":"sky","label":"Substitute","style":"solid"},{"to":"k-sub3","from":"k-val","color":"sky","label":"Substitute","style":"solid"}]},"order":2,"title":"Find the first three terms","description":"Now, we substitute $k = 12$ back into our expressions for the first three months to find the actual user numbers (in thousands)."},{"type":"text","order":3,"title":"Find the common ratio","content":"To find the common ratio $r$, we divide any term by its preceding term:\n\n$$r = \\frac{\\text{Term 2}}{\\text{Term 1}} = \\frac{12}{8} = 1.5$$\n\nLet's double check this with the next terms:\n\n$$r = \\frac{\\text{Term 3}}{\\text{Term 2}} = \\frac{18}{12} = 1.5$$\n\nSo, the common ratio is $r = 1.5$. Each month, the number of new premium users increases by a factor of $1.5$."},{"type":"text","order":4,"title":"Multiply sequentially to Month 6","content":"To find the number of new premium users joining in the 6th month, we can multiply step-by-step starting from Month 3:\n\n- **Month 3:** $18$ thousand\n- **Month 4:** $18 \\times 1.5 = 27$ thousand\n- **Month 5:** $27 \\times 1.5 = 40.5$ thousand\n- **Month 6:** $40.5 \\times 1.5 = 60.75$ thousand"},{"type":"text","order":5,"title":"State the final answer","content":"Since the terms are given in thousands, we convert our Month 6 value back into the actual number of users:\n\n$$60.75 \\times 1,000 = 60,750$$\n\nTherefore, the predicted number of new premium users joining in the 6th month is **$60,750$**."}]}],"generatedAt":"2026-06-09T08:05:25.252Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"b3bb30d1-85aa-41ba-83b9-c94256e3a632","label":"c","content":"The total number of all premium users who have joined the app since the update is predicted to exceed 5 million for the first time in the \$N\$th month.\n\nCalculate the value of \$N\$ according to the model.","markscheme":"- Identify that the sum $S_N$ must exceed 5000 (since 5 million is 5000 thousand) B1\n- Use $S_N = \\dfrac{a(r^N - 1)}{r - 1}$ to set up the inequality: $\\dfrac{8(1.5^N - 1)}{1.5 - 1} > 5000$ M1\n- Simplify to $16(1.5^N - 1) > 5000 \\Rightarrow 1.5^N > 313.5$ and apply logarithms: $N \\ln 1.5 > \\ln 313.5$ M1\n- Solve $N > \\dfrac{\\ln 313.5}{\\ln 1.5} \\approx 14.17$, so $N = 15$ A1","marks":4,"order":2,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"analytical_logarithms","label":"Analytical Approach using Logarithms","steps":[{"type":"text","order":0,"title":"Convert units and identify parameters","content":"First, we must align the units of our values. The question states that the number of new premium users is modeled in **thousands**.\n\nTherefore, we convert the target of 5 million premium users into thousands:\n\n$$\\text{Target} = 5,000,000 = 5000 \\text{ thousand}$$\n\nFrom the previous parts of this problem, we have the model parameters for the geometric series:\n- First term, $a = 8$\n- Common ratio, $r = 1.5$\n\nWe need to find the first month $N$ such that the sum of the first $N$ terms, $S_N$, exceeds $5000$ thousand:\n\n$$S_N > 5000$$","highlights":[{"text":"exceed 5 million","location":"specification"}]},{"type":"text","order":1,"title":"Set up the sum inequality","content":"Recall the formula for the sum of the first $N$ terms of a geometric series from the formula booklet:\n\n$$S_N = \\frac{a(r^N - 1)}{r - 1}$$\n\nSubstitute our parameters $a = 8$ and $r = 1.5$ into the sum formula and set it to be greater than $5000$:\n\n$$\\frac{8(1.5^N - 1)}{1.5 - 1} > 5000$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{8(1.5^N - 1)}{\\htmlData{anchor=denom}{\\color{@sky}{1.5 - 1}}} > 5000$"},{"latex":"$$\\frac{8(1.5^N - 1)}{\\htmlData{anchor=denom-val}{\\color{@sky}{0.5}}} > 5000$"},{"latex":"$$\\htmlData{anchor=coeff}{\\color{@amber}{16}}(1.5^N - 1) > 5000$"},{"latex":"$$1.5^N - 1 > \\frac{5000}{\\color{@amber}{16}}$"},{"latex":"$$1.5^N - 1 > \\htmlData{anchor=frac-val}{\\color{@purple}{312.5}}$"},{"latex":"$$1.5^N > \\htmlData{anchor=final-val}{\\color{@teal}{313.5}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"denom"},{"color":"amber","style":"text-color","anchor":"coeff"},{"color":"purple","style":"text-color","anchor":"frac-val"},{"color":"teal","style":"text-color","anchor":"final-val"}],"connections":[{"to":"denom-val","from":"denom","color":"sky","label":"simplify","style":"solid"},{"to":"coeff","from":"denom-val","color":"amber","label":"8 / 0.5 = 16","style":"solid"},{"to":"frac-val","from":"coeff","color":"purple","label":"divide","style":"solid"},{"to":"final-val","from":"frac-val","color":"teal","label":"+ 1","style":"solid"}]},"order":2,"title":"Simplify the inequality","description":"We simplify the inequality step-by-step by dividing out the common factor and adding 1 to both sides."},{"type":"text","order":3,"title":"Apply logarithms to solve for N","content":"To solve for $N$ analytically when it is in the exponent, we take the natural logarithm ($\n\\ln$) of both sides of the inequality:\n\n$$\\ln(1.5^N) > \\ln(313.5)$$\n\nUsing the power rule of logarithms, $\\ln(x^y) = y \\ln(x)$, we can bring the exponent $N$ to the front:\n\n$$N \\ln(1.5) > \\ln(313.5)$$\n\nSince $1.5 > 1$, $\\ln(1.5)$ is a positive number, so we can divide both sides by $\\ln(1.5)$ without reversing the inequality sign:\n\n$$N > \\frac{\\ln(313.5)}{\\ln(1.5)}$$"},{"type":"text","order":4,"title":"Calculate the final month N","content":"Now we evaluate the expression using a calculator:\n\n$$N > 14.172\\dots$$\n\nSince the month $N$ must be a whole number, we find the first integer value of $N$ that satisfies this inequality. \n\nThe first integer greater than $14.17$ is **$15$**.\n\nThus, the total number of premium users is predicted to exceed 5 million for the first time in the **$15$th** month.","calculator":[{"gdc":{"expression":"ln(313.5)/ln(1.5)","instructions":"Enter the logarithmic division formula directly into the run-matrix menu to find the decimal limit."},"ti84":{"expression":"ln(313.5)/ln(1.5)","instructions":"Press [LN], type 313.5, press [)], press [/], press [LN], type 1.5, press [)], then press [ENTER]."},"desmos":{"expression":"\\frac{\\ln(313.5)}{\\ln(1.5)}","instructions":"Type the expression in a blank cell to find the value of N."},"description":"Calculate the exact analytical limit for $N$ using logarithms."}]}]},{"id":"gdc_table","label":"Calculator Table or Graphical Solver","steps":[{"type":"text","order":0,"title":"Convert 5 million to thousands","content":"Since the number of new premium users is given in **thousands**, we first need to convert 5 million into thousands so that the units of our equation match. \n\n$$\\text{Target} = \\frac{5,000,000}{1,000} = 5000 \\text{ thousand}$$ \n\nWe want to find the smallest integer month $N$ such that the sum of new users up to month $N$ is strictly greater than $5000$:\n\n$$S_N > 5000$$","highlights":[{"text":"exceed 5 million","location":"specification"}]},{"type":"text","order":1,"title":"Set up the sum formula","content":"We recall the formula for the sum of the first $n$ terms of a geometric series from the formula booklet:\n\n$$S_n = \\frac{a(r^n - 1)}{r - 1}$$\n\nGiven the first term $a = 8$ and the common ratio $r = 1.5$, we substitute these values into the formula to express $S_N$:\n\n$$S_N = \\frac{8(1.5^N - 1)}{1.5 - 1}$$\n\nSimplifying the denominator $1.5 - 1 = 0.5$:\n\n$$S_N = \\frac{8(1.5^N - 1)}{0.5} = 16(1.5^N - 1)$$\n\nSo, our inequality is:\n\n$$16(1.5^N - 1) > 5000$$"},{"type":"text","order":2,"title":"Analyze using calculator table","content":"To find the first integer value of $N$ where $S_N > 5000$, we can use our graphing calculator's table feature to compute $S_N$ for successive values of $N$. Let's define the function:\n\n$$Y_1 = 16(1.5^x - 1)$$\n\nNow, generate the table of values to observe when $Y_1$ first exceeds $5000$.","calculator":[{"gdc":{"expression":"16*(1.5^x - 1)","instructions":"1. Go to the Graph/Table menu.\n2. Enter the function \\\$f1(x) = 16(1.5^x - 1)\\\$.\n3. Open the table of values (ctrl + T on TI-Nspire).\n4. Scroll through the x-values to find when the function value exceeds 5000."},"ti84":{"expression":"16*(1.5^X - 1)","instructions":"1. Press [Y=] and enter \\\$Y_1 = 16(1.5^X - 1)\\\$.\n2. Press [2nd] [WINDOW] (TBLSET). Set TblStart = 10 and \\\$\\Delta\\\$Tbl = 1.\n3. Press [2nd] [GRAPH] (TABLE) to view the table of values.\n4. Scroll down to find the first row where \\\$Y_1 > 5000\\\$."},"desmos":{"expression":"f(x) = 16*(1.5^x - 1)","instructions":"1. Type \\\$f(x) = 16(1.5^x - 1)\\\$ in the input bar.\n2. Click the gear icon (Tools) and select the \"Convert to table\" icon.\n3. Input values of x from 12 to 16 to inspect the corresponding outputs."},"description":"Generate a table of values for the sum function to find when the sum exceeds 5000."}]},{"type":"text","order":3,"title":"Evaluate table values","content":"Using the table of values, we find:\n\n- For $N = 13$:\n $$S_{13} = 16(1.5^{13} - 1) \\approx 16(194.62 - 1) \\approx 3097.9$$\n- For $N = 14$:\n $$S_{14} = 16(1.5^{14} - 1) \\approx 16(291.93 - 1) \\approx 4654.9$$\n- For $N = 15$:\n $$S_{15} = 16(1.5^{15} - 1) \\approx 16(437.89 - 1) \\approx 6990.3$$\n\nThus, the total predicted premium users exceed 5 million ($5000$ thousand) for the first time during month $15$."},{"type":"text","order":4,"title":"State final answer","content":"The first month in which the total number of premium users exceeds 5 million is **15**.\n\n$$N = 15$$"}]}],"generatedAt":"2026-06-09T08:06:11.884Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"7de09d2e-751b-428a-90d7-0caec45f59dd:common_ratio","label":"(a) Equating Common Ratios","steps":[{"type":"text","order":0,"title":"Equate the common ratios","content":"For three consecutive terms in a geometric sequence, the ratio between successive terms is constant. This is called the common ratio, $r$.\n\nWe can find two expressions for $r$ using our three terms, $3k - 28$, $k$, and $k + 6$:\n\n1. Divide the second term by the first term:\n$$r = \\frac{k}{3k - 28}$$\n\n2. Divide the third term by the second term:\n$$r = \\frac{k + 6}{k}$$\n\nSince the common ratio is the same, we can equate these two expressions:\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$"},{"type":"text","order":1,"title":"Cross-multiply and expand","content":"To eliminate the fractions, we cross-multiply both sides of the equation:\n\n$$k \\times k = (3k - 28)(k + 6)$$\n\n$$k^2 = (3k - 28)(k + 6)$$\n\nNow, expand the right-hand side using the FOIL method:\n\n$$k^2 = 3k^2 + 18k - 28k - 168$$\n\nSimplify the terms on the right-hand side:\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":2,"title":"Rearrange and simplify","content":"Rearrange the equation to set it equal to zero by subtracting $k^2$ from both sides:\n\n$$0 = 2k^2 - 10k - 168$$\n\nTo match the target equation, we can divide every term in the equation by $2$:\n\n$$0 = k^2 - 5k - 84$$\n\nRearranging gives us the desired quadratic equation:\n\n$$k^2 - 5k - 84 = 0$$"}]},{"id":"7de09d2e-751b-428a-90d7-0caec45f59dd:geometric_mean","label":"(a) Using Geometric Mean Relation","steps":[{"type":"text","order":0,"title":"Identify terms of the sequence","content":"We are given that the first three terms of a geometric series are:\n\n* $u_1 = 3k - 28$\n* $u_2 = k$\n* $u_3 = k + 6$\n\nwhere $k$ is a constant."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_1 = \\color{@sky}{3k - 28}, \\quad u_2 = \\color{@purple}{k}, \\quad u_3 = \\color{@amber}{k + 6}$"},{"latex":"$$(\\htmlData{anchor=u2-lhs}{\\color{@purple}{u_2}})^2 = \\htmlData{anchor=u1-rhs}{\\color{@sky}{u_1}} \\cdot \\htmlData{anchor=u3-rhs}{\\color{@amber}{u_3}}$"},{"latex":"$$(\\htmlData{anchor=k-lhs}{\\color{@purple}{k}})^2 = (\\htmlData{anchor=u1-sub}{\\color{@sky}{3k - 28}})(\\htmlData{anchor=u3-sub}{\\color{@amber}{k + 6}})$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"u2-lhs"},{"color":"sky","style":"text-color","anchor":"u1-rhs"},{"color":"amber","style":"text-color","anchor":"u3-rhs"}],"connections":[{"to":"k-lhs","from":"u2-lhs","color":"purple","label":"substitute","style":"solid"},{"to":"u1-sub","from":"u1-rhs","color":"sky","label":null,"style":"solid"},{"to":"u3-sub","from":"u3-rhs","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Apply geometric mean relation","description":"In any geometric progression, the ratio between consecutive terms is constant, which means the square of the middle term is equal to the product of the two outer terms: $u_2^2 = u_1 u_3$. This is called the geometric mean relation."},{"type":"text","order":2,"title":"Expand and simplify the equation","content":"Now, let's expand the right-hand side of the equation:\n\n$$(3k - 28)(k + 6) = 3k^2 + 18k - 28k - 168$$\n\nCombine the like terms in the expansion:\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":3,"title":"Rearrange to the final form","content":"To get our quadratic equation into standard form, we move all terms to one side:\n\n$$3k^2 - k^2 - 10k - 168 = 0$$\n\n$$2k^2 - 10k - 168 = 0$$\n\nNow, divide every term by $2$ to simplify to the final requested equation:\n\n$$k^2 - 5k - 84 = 0$$\n\nThis completes the proof as required."}]},{"id":"763732e7-bab0-4f62-a4cf-28baaedd9d5e:algebraic_formula","label":"(b) Algebraic Equation and Formula","steps":[{"type":"text","order":0,"title":"Set up the ratio equation","content":"For a sequence to be geometric, the ratio between consecutive terms must be constant. This constant is the common ratio, $r$.\n\nUsing the first three terms of our sequence, we can write:\n\n$$r = \\frac{u_2}{u_1} = \\frac{u_3}{u_2}$$\n\nSubstituting the terms given in the question, $u_1 = 3k - 28$, $u_2 = k$, and $u_3 = k + 6$, we get:\n\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$"},{"type":"text","order":1,"title":"Form a quadratic equation","content":"To solve this equation, cross-multiply to eliminate the fractions:\n\n$$k^2 = (k + 6)(3k - 28)$$\n\nExpand the right-hand side of the equation:\n\n$$k^2 = 3k^2 - 28k + 18k - 168$$\n\n$$k^2 = 3k^2 - 10k - 168$$\n\nRearrange everything to one side to form a standard quadratic equation:\n\n$$2k^2 - 10k - 168 = 0$$\n\nDividing the entire equation by $2$ simplifies it to:\n\n$$k^2 - 5k - 84 = 0$$"},{"type":"text","order":2,"title":"Solve for $k$","content":"Now we factorise the quadratic equation to find the values of $k$. We are looking for two numbers that multiply to $-84$ and add up to $-5$. These numbers are $-12$ and $+7$:\n\n$$(k - 12)(k + 7) = 0$$\n\nThis gives two possible solutions for $k$:\n\n$$k = 12 \\quad \\text{or} \\quad k = -7$$\n\nSince the number of new premium users must be positive, the term $u_2 = k$ must be positive. Therefore, we discard $k = -7$ and select:\n\n$$k = 12$$"},{"type":"text","order":3,"title":"Find the first term and ratio","content":"Now substitute $k = 12$ back to find the first term, $a$ (or $u_1$), and the common ratio, $r$:\n\n* First term: \n $$a = 3(12) - 28 = 36 - 28 = 8$$\n* Common ratio: \n $$r = \\frac{k}{a} = \\frac{12}{8} = 1.5$$"},{"type":"text","order":4,"title":"Calculate the 6th term","content":"The general formula for the $n$-th term of a geometric sequence is:\n\n$$u_n = a r^{n-1}$$\n\nTo find the number of users in the 6th month ($u_6$):\n\n$$u_6 = 8 \\times (1.5)^{6-1}$$\n\n$$u_6 = 8 \\times (1.5)^5$$\n\n$$u_6 = 8 \\times 7.59375 = 60.75$$"},{"type":"text","order":5,"title":"Convert to the final unit","content":"The question specifies that the number of premium users is tracked **in thousands**. \n\nTherefore, we multiply our result by $1,000$ to get the final prediction:\n\n$$\\text{New premium users} = 60.75 \\times 1,000 = 60,750$$"}]},{"id":"763732e7-bab0-4f62-a4cf-28baaedd9d5e:recursive_calculation","label":"(b) Step-by-Step Multiplication","steps":[{"type":"text","order":0,"title":"Set up the equation for $k$","content":"Since the number of new users joining each month forms a geometric sequence, the ratio between consecutive terms must be constant.\n\nThis means the common ratio $r$ can be written as:\n\n$$r = \\frac{\\text{Term 2}}{\\text{Term 1}} = \\frac{\\text{Term 3}}{\\text{Term 2}}$$\n\nSubstituting our terms $3k - 28$, $k$, and $k + 6$:\n\n$$\\frac{k}{3k - 28} = \\frac{k + 6}{k}$$\n\nNow, we cross-multiply to solve for $k$:\n\n$$k^2 = (k + 6)(3k - 28)$$\n\nExpanding the right-hand side:\n\n$$k^2 = 3k^2 - 28k + 18k - 168$$\n\n$$k^2 = 3k^2 - 10k - 168$$"},{"type":"text","order":1,"title":"Solve the quadratic equation","content":"Rearranging the equation to standard quadratic form $ax^2 + bx + c = 0$ by subtracting $k^2$ from both sides:\n\n$$2k^2 - 10k - 168 = 0$$\n\nDividing the entire equation by $2$ to simplify:\n\n$$k^2 - 5k - 84 = 0$$\n\nNow we factorise this quadratic. We need two numbers that multiply to $-84$ and add to $-5$. These numbers are $-12$ and $+7$:\n\n$$(k - 12)(k + 7) = 0$$\n\nThis gives two possible values for $k$:\n\n$$k = 12 \\quad \\text{or} \\quad k = -7$$\n\nSince the number of users joining must be positive (for example, Term 2 is $k$), we must choose the positive value:\n\n$$k = 12$$","calculator":[{"gdc":{"instructions":"Go to the Equation menu, select Polynomial (F2), choose Degree 2, enter coefficients 1, -5, and -84, then press SOLVE (F6)."},"ti84":{"instructions":"Press [APPS], select 'PlySmlt2' (Polynomial Root Finder and Simultaneous Equation Solver). Select 'Poly Root Finder', set order to 2, and enter coefficients: a = 1, b = -5, c = -84. Press [SOLVE]."},"desmos":{"expression":"y = x^2 - 5x - 84","instructions":"Graph the equation y = x^2 - 5x - 84 and identify the x-intercepts."},"description":"Verify the roots of the quadratic equation $k^2 - 5k - 84 = 0$ using a solver."}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$k = \\htmlData{anchor=k-val}{\\color{@sky}{12}}$"},{"latex":"$$\\text{Term 1: } 3(\\htmlData{anchor=k-sub1}{\\color{@sky}{12}}) - 28 = \\htmlData{anchor=t1}{\\color{@amber}{8}}$"},{"latex":"$$\\text{Term 2: } \\htmlData{anchor=t2}{\\color{@purple}{12}}$"},{"latex":"$$\\text{Term 3: } \\htmlData{anchor=k-sub3}{\\color{@sky}{12}} + 6 = \\htmlData{anchor=t3}{\\color{@green}{18}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"k-val"},{"color":"amber","style":"box","anchor":"t1"},{"color":"purple","style":"box","anchor":"t2"},{"color":"green","style":"box","anchor":"t3"}],"connections":[{"to":"k-sub1","from":"k-val","color":"sky","label":"Substitute","style":"solid"},{"to":"k-sub3","from":"k-val","color":"sky","label":"Substitute","style":"solid"}]},"order":2,"title":"Find the first three terms","description":"Now, we substitute $k = 12$ back into our expressions for the first three months to find the actual user numbers (in thousands)."},{"type":"text","order":3,"title":"Find the common ratio","content":"To find the common ratio $r$, we divide any term by its preceding term:\n\n$$r = \\frac{\\text{Term 2}}{\\text{Term 1}} = \\frac{12}{8} = 1.5$$\n\nLet's double check this with the next terms:\n\n$$r = \\frac{\\text{Term 3}}{\\text{Term 2}} = \\frac{18}{12} = 1.5$$\n\nSo, the common ratio is $r = 1.5$. Each month, the number of new premium users increases by a factor of $1.5$."},{"type":"text","order":4,"title":"Multiply sequentially to Month 6","content":"To find the number of new premium users joining in the 6th month, we can multiply step-by-step starting from Month 3:\n\n- **Month 3:** $18$ thousand\n- **Month 4:** $18 \\times 1.5 = 27$ thousand\n- **Month 5:** $27 \\times 1.5 = 40.5$ thousand\n- **Month 6:** $40.5 \\times 1.5 = 60.75$ thousand"},{"type":"text","order":5,"title":"State the final answer","content":"Since the terms are given in thousands, we convert our Month 6 value back into the actual number of users:\n\n$$60.75 \\times 1,000 = 60,750$$\n\nTherefore, the predicted number of new premium users joining in the 6th month is **$60,750$**."}]},{"id":"b3bb30d1-85aa-41ba-83b9-c94256e3a632:analytical_logarithms","label":"(c) Analytical Approach using Logarithms","steps":[{"type":"text","order":0,"title":"Convert units and identify parameters","content":"First, we must align the units of our values. The question states that the number of new premium users is modeled in **thousands**.\n\nTherefore, we convert the target of 5 million premium users into thousands:\n\n$$\\text{Target} = 5,000,000 = 5000 \\text{ thousand}$$\n\nFrom the previous parts of this problem, we have the model parameters for the geometric series:\n- First term, $a = 8$\n- Common ratio, $r = 1.5$\n\nWe need to find the first month $N$ such that the sum of the first $N$ terms, $S_N$, exceeds $5000$ thousand:\n\n$$S_N > 5000$$","highlights":[{"text":"exceed 5 million","location":"specification"}]},{"type":"text","order":1,"title":"Set up the sum inequality","content":"Recall the formula for the sum of the first $N$ terms of a geometric series from the formula booklet:\n\n$$S_N = \\frac{a(r^N - 1)}{r - 1}$$\n\nSubstitute our parameters $a = 8$ and $r = 1.5$ into the sum formula and set it to be greater than $5000$:\n\n$$\\frac{8(1.5^N - 1)}{1.5 - 1} > 5000$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{8(1.5^N - 1)}{\\htmlData{anchor=denom}{\\color{@sky}{1.5 - 1}}} > 5000$"},{"latex":"$$\\frac{8(1.5^N - 1)}{\\htmlData{anchor=denom-val}{\\color{@sky}{0.5}}} > 5000$"},{"latex":"$$\\htmlData{anchor=coeff}{\\color{@amber}{16}}(1.5^N - 1) > 5000$"},{"latex":"$$1.5^N - 1 > \\frac{5000}{\\color{@amber}{16}}$"},{"latex":"$$1.5^N - 1 > \\htmlData{anchor=frac-val}{\\color{@purple}{312.5}}$"},{"latex":"$$1.5^N > \\htmlData{anchor=final-val}{\\color{@teal}{313.5}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"denom"},{"color":"amber","style":"text-color","anchor":"coeff"},{"color":"purple","style":"text-color","anchor":"frac-val"},{"color":"teal","style":"text-color","anchor":"final-val"}],"connections":[{"to":"denom-val","from":"denom","color":"sky","label":"simplify","style":"solid"},{"to":"coeff","from":"denom-val","color":"amber","label":"8 / 0.5 = 16","style":"solid"},{"to":"frac-val","from":"coeff","color":"purple","label":"divide","style":"solid"},{"to":"final-val","from":"frac-val","color":"teal","label":"+ 1","style":"solid"}]},"order":2,"title":"Simplify the inequality","description":"We simplify the inequality step-by-step by dividing out the common factor and adding 1 to both sides."},{"type":"text","order":3,"title":"Apply logarithms to solve for N","content":"To solve for $N$ analytically when it is in the exponent, we take the natural logarithm ($\n\\ln$) of both sides of the inequality:\n\n$$\\ln(1.5^N) > \\ln(313.5)$$\n\nUsing the power rule of logarithms, $\\ln(x^y) = y \\ln(x)$, we can bring the exponent $N$ to the front:\n\n$$N \\ln(1.5) > \\ln(313.5)$$\n\nSince $1.5 > 1$, $\\ln(1.5)$ is a positive number, so we can divide both sides by $\\ln(1.5)$ without reversing the inequality sign:\n\n$$N > \\frac{\\ln(313.5)}{\\ln(1.5)}$$"},{"type":"text","order":4,"title":"Calculate the final month N","content":"Now we evaluate the expression using a calculator:\n\n$$N > 14.172\\dots$$\n\nSince the month $N$ must be a whole number, we find the first integer value of $N$ that satisfies this inequality. \n\nThe first integer greater than $14.17$ is **$15$**.\n\nThus, the total number of premium users is predicted to exceed 5 million for the first time in the **$15$th** month.","calculator":[{"gdc":{"expression":"ln(313.5)/ln(1.5)","instructions":"Enter the logarithmic division formula directly into the run-matrix menu to find the decimal limit."},"ti84":{"expression":"ln(313.5)/ln(1.5)","instructions":"Press [LN], type 313.5, press [)], press [/], press [LN], type 1.5, press [)], then press [ENTER]."},"desmos":{"expression":"\\frac{\\ln(313.5)}{\\ln(1.5)}","instructions":"Type the expression in a blank cell to find the value of N."},"description":"Calculate the exact analytical limit for $N$ using logarithms."}]}]},{"id":"b3bb30d1-85aa-41ba-83b9-c94256e3a632:gdc_table","label":"(c) Calculator Table or Graphical Solver","steps":[{"type":"text","order":0,"title":"Convert 5 million to thousands","content":"Since the number of new premium users is given in **thousands**, we first need to convert 5 million into thousands so that the units of our equation match. \n\n$$\\text{Target} = \\frac{5,000,000}{1,000} = 5000 \\text{ thousand}$$ \n\nWe want to find the smallest integer month $N$ such that the sum of new users up to month $N$ is strictly greater than $5000$:\n\n$$S_N > 5000$$","highlights":[{"text":"exceed 5 million","location":"specification"}]},{"type":"text","order":1,"title":"Set up the sum formula","content":"We recall the formula for the sum of the first $n$ terms of a geometric series from the formula booklet:\n\n$$S_n = \\frac{a(r^n - 1)}{r - 1}$$\n\nGiven the first term $a = 8$ and the common ratio $r = 1.5$, we substitute these values into the formula to express $S_N$:\n\n$$S_N = \\frac{8(1.5^N - 1)}{1.5 - 1}$$\n\nSimplifying the denominator $1.5 - 1 = 0.5$:\n\n$$S_N = \\frac{8(1.5^N - 1)}{0.5} = 16(1.5^N - 1)$$\n\nSo, our inequality is:\n\n$$16(1.5^N - 1) > 5000$$"},{"type":"text","order":2,"title":"Analyze using calculator table","content":"To find the first integer value of $N$ where $S_N > 5000$, we can use our graphing calculator's table feature to compute $S_N$ for successive values of $N$. Let's define the function:\n\n$$Y_1 = 16(1.5^x - 1)$$\n\nNow, generate the table of values to observe when $Y_1$ first exceeds $5000$.","calculator":[{"gdc":{"expression":"16*(1.5^x - 1)","instructions":"1. Go to the Graph/Table menu.\n2. Enter the function \\\$f1(x) = 16(1.5^x - 1)\\\$.\n3. Open the table of values (ctrl + T on TI-Nspire).\n4. Scroll through the x-values to find when the function value exceeds 5000."},"ti84":{"expression":"16*(1.5^X - 1)","instructions":"1. Press [Y=] and enter \\\$Y_1 = 16(1.5^X - 1)\\\$.\n2. Press [2nd] [WINDOW] (TBLSET). Set TblStart = 10 and \\\$\\Delta\\\$Tbl = 1.\n3. Press [2nd] [GRAPH] (TABLE) to view the table of values.\n4. Scroll down to find the first row where \\\$Y_1 > 5000\\\$."},"desmos":{"expression":"f(x) = 16*(1.5^x - 1)","instructions":"1. Type \\\$f(x) = 16(1.5^x - 1)\\\$ in the input bar.\n2. Click the gear icon (Tools) and select the \"Convert to table\" icon.\n3. Input values of x from 12 to 16 to inspect the corresponding outputs."},"description":"Generate a table of values for the sum function to find when the sum exceeds 5000."}]},{"type":"text","order":3,"title":"Evaluate table values","content":"Using the table of values, we find:\n\n- For $N = 13$:\n $$S_{13} = 16(1.5^{13} - 1) \\approx 16(194.62 - 1) \\approx 3097.9$$\n- For $N = 14$:\n $$S_{14} = 16(1.5^{14} - 1) \\approx 16(291.93 - 1) \\approx 4654.9$$\n- For $N = 15$:\n $$S_{15} = 16(1.5^{15} - 1) \\approx 16(437.89 - 1) \\approx 6990.3$$\n\nThus, the total predicted premium users exceed 5 million ($5000$ thousand) for the first time during month $15$."},{"type":"text","order":4,"title":"State final answer","content":"The first month in which the total number of premium users exceeds 5 million is **15**.\n\n$$N = 15$$"}]}],"generatedAt":"2026-06-09T08:06:23.989Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"9aadfd8a-f1df-4d17-a73d-c82be4a32f66","revisionId":"91883728-ba11-46b5-a1e6-1125d2357938","questionNumber":null,"questionType":"LA","specification":"The arithmetic series \$S\$ is given by\n\n\\[S = 7 + 11 + 15 + 19 + \\cdots + 303\\]\n\nFind","difficulty":3,"difficultyBand":"medium","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"21372bfe-f35c-4952-b631-215dba2e31d3","label":"a","content":"the number of terms in the series,","markscheme":"- Identifies the first term $a = 7$, common difference $d = 4$ and last term $l = 303$ B1\n- Uses the formula $u_n = a + (n - 1)d$ to set up the equation $303 = 7 + (n - 1)4$ M1\n- Solves the equation to find $n = 75$ A1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"formulaic","label":"Using the n-th Term Formula","steps":[{"type":"text","order":0,"title":"Identify key parameters","content":"To find the number of terms in the arithmetic series, we first need to identify the key components of the sequence:\n\n* **First term ($a$):** The first number in the series is $7$.\n* **Common difference ($d$):** The difference between consecutive terms is $11 - 7 = 4$.\n* **Last term ($u_n$ or $l$):** The final term of the series is $303$.\n\nIdentifying these values correctly earns the first mark B1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_n = \\htmlData{anchor=a-term}{\\color{@sky}{a}} + (n - 1)\\htmlData{anchor=d-diff}{\\color{@amber}{d}}$"},{"latex":"$$\\htmlData{anchor=un-term}{\\color{@purple}{303}} = \\htmlData{anchor=a-sub}{\\color{@sky}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"d-diff"},{"color":"purple","style":"text-color","anchor":"un-term"}],"connections":[{"to":"a-sub","from":"a-term","color":"sky","label":"first term","style":"solid"},{"to":"d-sub","from":"d-diff","color":"amber","label":"common difference","style":"solid"}]},"order":1,"title":"Set up the equation","description":"We substitute our identified values into the $n$-th term formula for an arithmetic progression. This substitution earns a method mark M1."},{"type":"text","order":2,"title":"Solve for n","content":"Now we solve the equation for $n$ step-by-step:\n\n$$303 = 7 + 4(n - 1)$$\n\n$$296 = 4(n - 1)$$\n\n$$74 = n - 1$$\n\n$$n = 75$$\n\nThus, there are $75$ terms in the series. Finding this value correctly earns the final mark A1.","calculator":[{"gdc":{"expression":"303 = 7 + (X - 1) * 4","instructions":"Go to the Equation menu, select Solver, enter the equation `303 = 7 + (X - 1) * 4`, and solve for $X$."},"ti84":{"expression":"303 = 7 + (N - 1) * 4","instructions":"Press [MATH], scroll down to [Numeric Solver...]. Enter `E1: 303` and `E2: 7 + (N - 1) * 4`, then press [OK] (graph button) and [SOLVE] to find $N$."},"desmos":{"expression":"303 = 7 + (x - 1) * 4","instructions":"Type the equation in the input bar to find the vertical line representing the root."},"description":"You can verify this result using your graphing calculator's equation solver or by graphing."}]},{"type":"text","order":3,"title":"Calculate the sum","content":"Now we find the sum of the series using the sum formula:\n\n$$S_n = \\frac{n}{2}(a + l)$$\n\nSubstituting $n = 75$, $a = 7$, and $l = 303$ into the formula:\n\n$$S_{75} = \\frac{75}{2}(7 + 303)$$\n\n$$S_{75} = \\frac{75}{2}(310)$$\n\n$$S_{75} = 11\\,625$$\n\nThis substitution earns the method marks M1 M1 and the final accuracy mark A1."}]},{"id":"intuitive","label":"Step-Counting Approach","steps":[{"type":"text","order":0,"title":"Identify key terms","content":"First, let's identify the first term, the last term, and the common difference from the given arithmetic series:\n\n$$S = 7 + 11 + 15 + 19 + \\cdots + 303$$\n\n* **First term ($a$):** $7$\n* **Last term ($l$):** $303$\n* **Common difference ($d$):** $11 - 7 = 4$\n\n*This earns you your first mark **[B1]**.*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Total Difference} = \\htmlData{anchor=last-term}{\\color{@sky}{303}} - \\htmlData{anchor=first-term}{\\color{@amber}{7}} = 296$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"last-term"},{"color":"amber","style":"text-color","anchor":"first-term"}],"connections":[]},"order":1,"title":"Find the total difference","description":"We calculate the total difference between the last term and the first term to see how much the sequence grows overall."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Number of Steps} = \\frac{\\htmlData{anchor=total-diff}{\\color{@teal}{296}}}{\\htmlData{anchor=common-diff}{\\color{@purple}{4}}} = 74$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"total-diff"},{"color":"purple","style":"text-color","anchor":"common-diff"}],"connections":[]},"order":2,"title":"Calculate the steps","description":"Next, divide the total difference by the common difference to find how many steps (or jumps of $4$) are taken to get from the first term to the last term."},{"type":"text","order":3,"title":"Determine total terms","content":"Since we started at the first term, the number of terms is always **one more** than the number of steps:\n\n$$\\text{Number of terms } (n) = \\text{Number of steps} + 1$$\n\n$$n = 74 + 1$$\n\n$$n = 75$$\n\nThis calculation corresponds to solving the standard formula $l = a + (n - 1)d$, which earns the method mark **[M1]** and the final accuracy mark **[A1]**.\n\nTherefore, there are **$75$** terms in the series.","calculator":[{"gdc":{"expression":"(303-7)/4+1","instructions":"Type (303 - 7) / 4 + 1 and press ENTER."},"ti84":{"expression":"(303-7)/4+1","instructions":"Type (303 - 7) / 4 + 1 and press ENTER."},"desmos":{"expression":"\\frac{303-7}{4}+1","instructions":"Enter the calculation into the input field to get the result."},"description":"You can quickly verify the arithmetic on your calculator."}]},{"type":"text","order":4,"title":"Calculate the sum","content":"With the number of terms known, we can calculate the total sum using the alternative arithmetic sum formula:\n\n$$S_n = \\frac{n}{2}[2a + (n - 1)d]$$\n\nSubstituting $n = 75$, $a = 7$, and $d = 4$:\n\n$$S_{75} = \\frac{75}{2}[2(7) + (75 - 1)4]$$\n\n$$S_{75} = \\frac{75}{2}[14 + 74 \\times 4]$$\n\n$$S_{75} = \\frac{75}{2}[310]$$\n\n$$S_{75} = 11\\,625$$\n\nThis substitution earns **[M1]** and the final sum of **$11\\,625$** earns **[A1]**."}]}],"generatedAt":"2026-06-09T07:59:13.284Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"36eba81c-718b-4b89-8bde-80a4d824f523","label":"b","content":"the sum of the series.","markscheme":"- Uses the arithmetic series sum formula $S_n = \\frac{n}{2}(a + l)$ or $S_n = \\frac{n}{2}(2a + (n - 1)d)$ M1\n- Substitutes $n = 75$, $a = 7$, and $l = 303$ into the formula to get $S_{75} = \\frac{75}{2}(7 + 303)$ M1\n- Obtains the final sum $11\\,625$ A1","marks":3,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"first_last_term_formula","label":"Using First and Last Term Formula","steps":[{"type":"text","order":0,"title":"Identify series parameters","content":"To sum the arithmetic series $S = 7 + 11 + 15 + 19 + \\dots + 303$, we first identify its key components:\n\n- **First term** ($a$) = $7$\n- **Common difference** ($d$) = $11 - 7 = 4$\n- **Last term** ($l$) = $303$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Use the } n\\text{-th term formula: } \\htmlData{anchor=nth-form}{\\color{@sky}{u_n = a + (n - 1)d}}$"},{"latex":"$$\\htmlData{anchor=sub-eq}{\\color{@sky}{303}} = \\htmlData{anchor=a-sub}{\\color{@purple}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"},{"latex":"$$296 = 4(n - 1)$"},{"latex":"$$n - 1 = 74 \\implies \\htmlData{anchor=n-val}{\\color{@green}{n = 75}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-eq"},{"color":"purple","style":"text-color","anchor":"a-sub"},{"color":"amber","style":"text-color","anchor":"d-sub"},{"color":"green","style":"box","anchor":"n-val"}],"connections":[{"to":"sub-eq","from":"nth-form","color":"sky","label":"Substitute values","style":"solid"}]},"order":1,"title":"Find the number of terms","description":"We need to determine the total number of terms, $n$, in the series. We use the formula for the $n$-th term of an arithmetic progression, $u_n = a + (n - 1)d$, setting $u_n = 303$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Arithmetic series sum formula: } \\htmlData{anchor=sum-formula}{\\color{@teal}{S_n = \\frac{n}{2}(a + l)}}$"},{"latex":"$$\\htmlData{anchor=sum-sub}{S_{75}} = \\frac{\\htmlData{anchor=n-sub}{\\color{@green}{75}}}{2}\\left(\\htmlData{anchor=a-val}{\\color{@purple}{7}} + \\htmlData{anchor=l-val}{\\color{@sky}{303}}\\right)$"},{"latex":"$$S_{75} = 37.5 \\times 310$"},{"latex":"$$S_{75} = \\htmlData{anchor=final-ans}{\\color{@orange}{11\\,625}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"n-sub"},{"color":"purple","style":"text-color","anchor":"a-val"},{"color":"sky","style":"text-color","anchor":"l-val"},{"color":"orange","style":"box","anchor":"final-ans"}],"connections":[{"to":"sum-sub","from":"sum-formula","color":"teal","label":"Substitute n, a, l","style":"solid"}]},"order":2,"title":"Apply sum formula","description":"Now that we know there are $75$ terms, we can find the sum using the formula $S_n = \\frac{n}{2}(a + l)$, where $l$ is the last term."},{"type":"text","order":3,"title":"Verify with GDC","content":"You can verify this result by evaluating the sum directly using sigma notation on your calculator.","calculator":[{"gdc":{"expression":"\\sum_{X=1}^{75} (7 + (X-1) \\cdot 4)","instructions":"Go to the Run-Matrix menu, press MATH, then Σ(. Input variable X, from 1 to 75, and the expression 7 + (X - 1) × 4."},"ti84":{"expression":"\\sum_{X=1}^{75} (7 + (X-1) \\cdot 4)","instructions":"Press ALPHA then WINDOW (F2), select 2: summation Σ(. Enter the variable X, starting value 1, end value 75, and the formula 7 + (X - 1)*4."},"desmos":{"expression":"\\sum_{n=1}^{75} (7 + (n-1) \\cdot 4)","instructions":"Type sum in an empty expression bar to generate a summation template. Set the limits from n=1 to 75 and input the expression 7 + (n-1)*4."},"description":"Calculate the sum of the series using the sum template."}]}]},{"id":"common_difference_formula","label":"Using Common Difference Formula","steps":[{"type":"text","order":0,"title":"Identify series parameters","content":"From the given arithmetic series:\n\n$$S = 7 + 11 + 15 + 19 + \\dots + 303$$\n\nWe can identify the key parameters:\n\n- First term ($a$) = $7$\n- Common difference ($d$) = $11 - 7 = 4$\n- Last term ($u_n$ or $l$) = $303$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_n = \\htmlData{anchor=a-term}{\\color{@sky}{a}} + (n - 1)\\htmlData{anchor=d-diff}{\\color{@amber}{d}}$"},{"latex":"$$303 = \\htmlData{anchor=a-sub}{\\color{@sky}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"},{"latex":"$$296 = 4(n - 1) \\implies n - 1 = 74 \\implies n = 75$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"d-diff"}],"connections":[{"to":"a-sub","from":"a-term","color":"sky","label":"substitute a = 7","style":"solid"},{"to":"d-sub","from":"d-diff","color":"amber","label":"substitute d = 4","style":"solid"}]},"order":1,"title":"Find the number of terms","description":"We use the general $n^{\\text{th}}$ term formula for an arithmetic sequence to solve for $n$, the total number of terms in the series."},{"type":"text","order":2,"title":"Calculate the sum","content":"Now we substitute our values ($n = 75$, $a = 7$, and $d = 4$) into the arithmetic series sum formula involving the common difference:\n\n$$S_n = \\frac{n}{2}\\left(2a + (n - 1)d\\right)$$\n\n$$\\begin{aligned} S_{75} &= \\frac{75}{2}\\left(2(7) + (75 - 1)4\\right) \\\\ &= \\frac{75}{2}\\left(14 + (74)4\\right) \\\\ &= \\frac{75}{2}\\left(14 + 296\\right) \\\\ &= \\frac{75}{2}(310) \\\\ &= 75 \\times 155 \\\\ &= 11\\,625 \\end{aligned}$$\n\nThis earns the method marks M1 M1 and accuracy mark A1.","calculator":[{"gdc":{"expression":"75 / 2 * (2 * 7 + 74 * 4)","instructions":"Enter '75 / 2 * (2 * 7 + 74 * 4)' on the run screen and execute."},"ti84":{"expression":"75 / 2 * (2 * 7 + 74 * 4)","instructions":"Type the arithmetic expression on the home screen: '75 / 2 * (2 * 7 + 74 * 4)' and press ENTER."},"desmos":{"expression":"\\frac{75}{2}\\left(2\\cdot7+\\left(75-1\\right)\\cdot4\\right)","instructions":"Type the formula directly to evaluate the sum."},"description":"Verify the calculation on your graphing calculator."}]}]},{"id":"calculator_sigma_notation","label":"Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Analyze the series","content":"The given arithmetic series is:\n\n$$S = 7 + 11 + 15 + 19 + \\dots + 303$$\n\nTo represent this series in sigma notation, we first find the general term ($r$-th term), $u_r$.\n\n- The first term is $a = 7$.\n- The common difference is $d = 11 - 7 = 4$.\n\nThe formula for the $r$-th term of an arithmetic sequence is:\n\n$$u_r = a + (r - 1)d$$\n\nSubstituting our values in:\n\n$$u_r = 7 + (r - 1)4 = 4r + 3$$"},{"type":"text","order":1,"title":"Find number of terms","content":"Next, we find the number of terms, $n$, by setting the general term equal to the last term, $303$:\n\n$$4n + 3 = 303$$\n\nSolving for $n$:\n\n$$\\begin{aligned} 4n &= 300 \\\\ n &= 75 \\end{aligned}$$\n\nSo, there are $75$ terms in this series."},{"type":"text","order":2,"title":"Write in sigma notation","content":"We can now write the sum of the series using sigma notation as:\n\n$$S = \\sum_{r=1}^{75} (4r + 3)$$\n\nWe can enter this expression directly into a graphing calculator to find the final sum."},{"type":"text","order":3,"title":"Evaluate with calculator","content":"Inputting the summation into your graphing calculator will evaluate the total sum of the series directly.","calculator":[{"gdc":{"expression":"\\sum_{X=1}^{75} (4X+3)","instructions":"Go to the Run-Matrix menu. Press [MATH] then [F6] to see more options, and select Sigma ([F2]).\nFill in the template:\n- Enter the expression: 4X+3\n- Set the variable: X\n- Set the start value: 1\n- Set the end value: 75\nPress [EXE] to calculate."},"ti84":{"expression":"sum(seq(4X+3,X,1,75))","instructions":"Press [ALPHA] [WINDOW] and select 2:summation(\nOr press [MATH] and scroll down to 0:summation(\nFill in the template:\n- Under the Sigma symbol, enter X=1\n- On top of the Sigma symbol, enter 75\n- In the parentheses, enter 4X+3\nPress [ENTER] to calculate."},"desmos":{"expression":"\\sum_{n=1}^{75} (4n+3)","instructions":"Type 'sum' in an expression line to bring up the summation notation.\nSet the lower limit to n=1, the upper limit to 75, and enter 4n+3 inside the summation."},"description":"Evaluate the sum of 4r + 3 from r = 1 to 75."}]},{"type":"text","order":4,"title":"State final answer","content":"The graphing calculator outputs the final sum:\n\n$$S = 11\\,625$$\n\nThis confirms the analytical calculation of the arithmetic series sum."}]}],"generatedAt":"2026-06-09T08:00:56.446Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"21372bfe-f35c-4952-b631-215dba2e31d3:formulaic","label":"(a) Using the n-th Term Formula","steps":[{"type":"text","order":0,"title":"Identify key parameters","content":"To find the number of terms in the arithmetic series, we first need to identify the key components of the sequence:\n\n* **First term ($a$):** The first number in the series is $7$.\n* **Common difference ($d$):** The difference between consecutive terms is $11 - 7 = 4$.\n* **Last term ($u_n$ or $l$):** The final term of the series is $303$.\n\nIdentifying these values correctly earns the first mark B1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_n = \\htmlData{anchor=a-term}{\\color{@sky}{a}} + (n - 1)\\htmlData{anchor=d-diff}{\\color{@amber}{d}}$"},{"latex":"$$\\htmlData{anchor=un-term}{\\color{@purple}{303}} = \\htmlData{anchor=a-sub}{\\color{@sky}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"d-diff"},{"color":"purple","style":"text-color","anchor":"un-term"}],"connections":[{"to":"a-sub","from":"a-term","color":"sky","label":"first term","style":"solid"},{"to":"d-sub","from":"d-diff","color":"amber","label":"common difference","style":"solid"}]},"order":1,"title":"Set up the equation","description":"We substitute our identified values into the $n$-th term formula for an arithmetic progression. This substitution earns a method mark M1."},{"type":"text","order":2,"title":"Solve for n","content":"Now we solve the equation for $n$ step-by-step:\n\n$$303 = 7 + 4(n - 1)$$\n\n$$296 = 4(n - 1)$$\n\n$$74 = n - 1$$\n\n$$n = 75$$\n\nThus, there are $75$ terms in the series. Finding this value correctly earns the final mark A1.","calculator":[{"gdc":{"expression":"303 = 7 + (X - 1) * 4","instructions":"Go to the Equation menu, select Solver, enter the equation `303 = 7 + (X - 1) * 4`, and solve for $X$."},"ti84":{"expression":"303 = 7 + (N - 1) * 4","instructions":"Press [MATH], scroll down to [Numeric Solver...]. Enter `E1: 303` and `E2: 7 + (N - 1) * 4`, then press [OK] (graph button) and [SOLVE] to find $N$."},"desmos":{"expression":"303 = 7 + (x - 1) * 4","instructions":"Type the equation in the input bar to find the vertical line representing the root."},"description":"You can verify this result using your graphing calculator's equation solver or by graphing."}]},{"type":"text","order":3,"title":"Calculate the sum","content":"Now we find the sum of the series using the sum formula:\n\n$$S_n = \\frac{n}{2}(a + l)$$\n\nSubstituting $n = 75$, $a = 7$, and $l = 303$ into the formula:\n\n$$S_{75} = \\frac{75}{2}(7 + 303)$$\n\n$$S_{75} = \\frac{75}{2}(310)$$\n\n$$S_{75} = 11\\,625$$\n\nThis substitution earns the method marks M1 M1 and the final accuracy mark A1."}]},{"id":"21372bfe-f35c-4952-b631-215dba2e31d3:intuitive","label":"(a) Step-Counting Approach","steps":[{"type":"text","order":0,"title":"Identify key terms","content":"First, let's identify the first term, the last term, and the common difference from the given arithmetic series:\n\n$$S = 7 + 11 + 15 + 19 + \\cdots + 303$$\n\n* **First term ($a$):** $7$\n* **Last term ($l$):** $303$\n* **Common difference ($d$):** $11 - 7 = 4$\n\n*This earns you your first mark **[B1]**.*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Total Difference} = \\htmlData{anchor=last-term}{\\color{@sky}{303}} - \\htmlData{anchor=first-term}{\\color{@amber}{7}} = 296$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"last-term"},{"color":"amber","style":"text-color","anchor":"first-term"}],"connections":[]},"order":1,"title":"Find the total difference","description":"We calculate the total difference between the last term and the first term to see how much the sequence grows overall."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Number of Steps} = \\frac{\\htmlData{anchor=total-diff}{\\color{@teal}{296}}}{\\htmlData{anchor=common-diff}{\\color{@purple}{4}}} = 74$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"total-diff"},{"color":"purple","style":"text-color","anchor":"common-diff"}],"connections":[]},"order":2,"title":"Calculate the steps","description":"Next, divide the total difference by the common difference to find how many steps (or jumps of $4$) are taken to get from the first term to the last term."},{"type":"text","order":3,"title":"Determine total terms","content":"Since we started at the first term, the number of terms is always **one more** than the number of steps:\n\n$$\\text{Number of terms } (n) = \\text{Number of steps} + 1$$\n\n$$n = 74 + 1$$\n\n$$n = 75$$\n\nThis calculation corresponds to solving the standard formula $l = a + (n - 1)d$, which earns the method mark **[M1]** and the final accuracy mark **[A1]**.\n\nTherefore, there are **$75$** terms in the series.","calculator":[{"gdc":{"expression":"(303-7)/4+1","instructions":"Type (303 - 7) / 4 + 1 and press ENTER."},"ti84":{"expression":"(303-7)/4+1","instructions":"Type (303 - 7) / 4 + 1 and press ENTER."},"desmos":{"expression":"\\frac{303-7}{4}+1","instructions":"Enter the calculation into the input field to get the result."},"description":"You can quickly verify the arithmetic on your calculator."}]},{"type":"text","order":4,"title":"Calculate the sum","content":"With the number of terms known, we can calculate the total sum using the alternative arithmetic sum formula:\n\n$$S_n = \\frac{n}{2}[2a + (n - 1)d]$$\n\nSubstituting $n = 75$, $a = 7$, and $d = 4$:\n\n$$S_{75} = \\frac{75}{2}[2(7) + (75 - 1)4]$$\n\n$$S_{75} = \\frac{75}{2}[14 + 74 \\times 4]$$\n\n$$S_{75} = \\frac{75}{2}[310]$$\n\n$$S_{75} = 11\\,625$$\n\nThis substitution earns **[M1]** and the final sum of **$11\\,625$** earns **[A1]**."}]},{"id":"36eba81c-718b-4b89-8bde-80a4d824f523:first_last_term_formula","label":"(b) Using First and Last Term Formula","steps":[{"type":"text","order":0,"title":"Identify series parameters","content":"To sum the arithmetic series $S = 7 + 11 + 15 + 19 + \\dots + 303$, we first identify its key components:\n\n- **First term** ($a$) = $7$\n- **Common difference** ($d$) = $11 - 7 = 4$\n- **Last term** ($l$) = $303$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Use the } n\\text{-th term formula: } \\htmlData{anchor=nth-form}{\\color{@sky}{u_n = a + (n - 1)d}}$"},{"latex":"$$\\htmlData{anchor=sub-eq}{\\color{@sky}{303}} = \\htmlData{anchor=a-sub}{\\color{@purple}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"},{"latex":"$$296 = 4(n - 1)$"},{"latex":"$$n - 1 = 74 \\implies \\htmlData{anchor=n-val}{\\color{@green}{n = 75}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-eq"},{"color":"purple","style":"text-color","anchor":"a-sub"},{"color":"amber","style":"text-color","anchor":"d-sub"},{"color":"green","style":"box","anchor":"n-val"}],"connections":[{"to":"sub-eq","from":"nth-form","color":"sky","label":"Substitute values","style":"solid"}]},"order":1,"title":"Find the number of terms","description":"We need to determine the total number of terms, $n$, in the series. We use the formula for the $n$-th term of an arithmetic progression, $u_n = a + (n - 1)d$, setting $u_n = 303$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Arithmetic series sum formula: } \\htmlData{anchor=sum-formula}{\\color{@teal}{S_n = \\frac{n}{2}(a + l)}}$"},{"latex":"$$\\htmlData{anchor=sum-sub}{S_{75}} = \\frac{\\htmlData{anchor=n-sub}{\\color{@green}{75}}}{2}\\left(\\htmlData{anchor=a-val}{\\color{@purple}{7}} + \\htmlData{anchor=l-val}{\\color{@sky}{303}}\\right)$"},{"latex":"$$S_{75} = 37.5 \\times 310$"},{"latex":"$$S_{75} = \\htmlData{anchor=final-ans}{\\color{@orange}{11\\,625}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"n-sub"},{"color":"purple","style":"text-color","anchor":"a-val"},{"color":"sky","style":"text-color","anchor":"l-val"},{"color":"orange","style":"box","anchor":"final-ans"}],"connections":[{"to":"sum-sub","from":"sum-formula","color":"teal","label":"Substitute n, a, l","style":"solid"}]},"order":2,"title":"Apply sum formula","description":"Now that we know there are $75$ terms, we can find the sum using the formula $S_n = \\frac{n}{2}(a + l)$, where $l$ is the last term."},{"type":"text","order":3,"title":"Verify with GDC","content":"You can verify this result by evaluating the sum directly using sigma notation on your calculator.","calculator":[{"gdc":{"expression":"\\sum_{X=1}^{75} (7 + (X-1) \\cdot 4)","instructions":"Go to the Run-Matrix menu, press MATH, then Σ(. Input variable X, from 1 to 75, and the expression 7 + (X - 1) × 4."},"ti84":{"expression":"\\sum_{X=1}^{75} (7 + (X-1) \\cdot 4)","instructions":"Press ALPHA then WINDOW (F2), select 2: summation Σ(. Enter the variable X, starting value 1, end value 75, and the formula 7 + (X - 1)*4."},"desmos":{"expression":"\\sum_{n=1}^{75} (7 + (n-1) \\cdot 4)","instructions":"Type sum in an empty expression bar to generate a summation template. Set the limits from n=1 to 75 and input the expression 7 + (n-1)*4."},"description":"Calculate the sum of the series using the sum template."}]}]},{"id":"36eba81c-718b-4b89-8bde-80a4d824f523:common_difference_formula","label":"(b) Using Common Difference Formula","steps":[{"type":"text","order":0,"title":"Identify series parameters","content":"From the given arithmetic series:\n\n$$S = 7 + 11 + 15 + 19 + \\dots + 303$$\n\nWe can identify the key parameters:\n\n- First term ($a$) = $7$\n- Common difference ($d$) = $11 - 7 = 4$\n- Last term ($u_n$ or $l$) = $303$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u_n = \\htmlData{anchor=a-term}{\\color{@sky}{a}} + (n - 1)\\htmlData{anchor=d-diff}{\\color{@amber}{d}}$"},{"latex":"$$303 = \\htmlData{anchor=a-sub}{\\color{@sky}{7}} + (n - 1)\\htmlData{anchor=d-sub}{\\color{@amber}{4}}$"},{"latex":"$$296 = 4(n - 1) \\implies n - 1 = 74 \\implies n = 75$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"d-diff"}],"connections":[{"to":"a-sub","from":"a-term","color":"sky","label":"substitute a = 7","style":"solid"},{"to":"d-sub","from":"d-diff","color":"amber","label":"substitute d = 4","style":"solid"}]},"order":1,"title":"Find the number of terms","description":"We use the general $n^{\\text{th}}$ term formula for an arithmetic sequence to solve for $n$, the total number of terms in the series."},{"type":"text","order":2,"title":"Calculate the sum","content":"Now we substitute our values ($n = 75$, $a = 7$, and $d = 4$) into the arithmetic series sum formula involving the common difference:\n\n$$S_n = \\frac{n}{2}\\left(2a + (n - 1)d\\right)$$\n\n$$\\begin{aligned} S_{75} &= \\frac{75}{2}\\left(2(7) + (75 - 1)4\\right) \\\\ &= \\frac{75}{2}\\left(14 + (74)4\\right) \\\\ &= \\frac{75}{2}\\left(14 + 296\\right) \\\\ &= \\frac{75}{2}(310) \\\\ &= 75 \\times 155 \\\\ &= 11\\,625 \\end{aligned}$$\n\nThis earns the method marks M1 M1 and accuracy mark A1.","calculator":[{"gdc":{"expression":"75 / 2 * (2 * 7 + 74 * 4)","instructions":"Enter '75 / 2 * (2 * 7 + 74 * 4)' on the run screen and execute."},"ti84":{"expression":"75 / 2 * (2 * 7 + 74 * 4)","instructions":"Type the arithmetic expression on the home screen: '75 / 2 * (2 * 7 + 74 * 4)' and press ENTER."},"desmos":{"expression":"\\frac{75}{2}\\left(2\\cdot7+\\left(75-1\\right)\\cdot4\\right)","instructions":"Type the formula directly to evaluate the sum."},"description":"Verify the calculation on your graphing calculator."}]}]},{"id":"36eba81c-718b-4b89-8bde-80a4d824f523:calculator_sigma_notation","label":"(b) Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Analyze the series","content":"The given arithmetic series is:\n\n$$S = 7 + 11 + 15 + 19 + \\dots + 303$$\n\nTo represent this series in sigma notation, we first find the general term ($r$-th term), $u_r$.\n\n- The first term is $a = 7$.\n- The common difference is $d = 11 - 7 = 4$.\n\nThe formula for the $r$-th term of an arithmetic sequence is:\n\n$$u_r = a + (r - 1)d$$\n\nSubstituting our values in:\n\n$$u_r = 7 + (r - 1)4 = 4r + 3$$"},{"type":"text","order":1,"title":"Find number of terms","content":"Next, we find the number of terms, $n$, by setting the general term equal to the last term, $303$:\n\n$$4n + 3 = 303$$\n\nSolving for $n$:\n\n$$\\begin{aligned} 4n &= 300 \\\\ n &= 75 \\end{aligned}$$\n\nSo, there are $75$ terms in this series."},{"type":"text","order":2,"title":"Write in sigma notation","content":"We can now write the sum of the series using sigma notation as:\n\n$$S = \\sum_{r=1}^{75} (4r + 3)$$\n\nWe can enter this expression directly into a graphing calculator to find the final sum."},{"type":"text","order":3,"title":"Evaluate with calculator","content":"Inputting the summation into your graphing calculator will evaluate the total sum of the series directly.","calculator":[{"gdc":{"expression":"\\sum_{X=1}^{75} (4X+3)","instructions":"Go to the Run-Matrix menu. Press [MATH] then [F6] to see more options, and select Sigma ([F2]).\nFill in the template:\n- Enter the expression: 4X+3\n- Set the variable: X\n- Set the start value: 1\n- Set the end value: 75\nPress [EXE] to calculate."},"ti84":{"expression":"sum(seq(4X+3,X,1,75))","instructions":"Press [ALPHA] [WINDOW] and select 2:summation(\nOr press [MATH] and scroll down to 0:summation(\nFill in the template:\n- Under the Sigma symbol, enter X=1\n- On top of the Sigma symbol, enter 75\n- In the parentheses, enter 4X+3\nPress [ENTER] to calculate."},"desmos":{"expression":"\\sum_{n=1}^{75} (4n+3)","instructions":"Type 'sum' in an expression line to bring up the summation notation.\nSet the lower limit to n=1, the upper limit to 75, and enter 4n+3 inside the summation."},"description":"Evaluate the sum of 4r + 3 from r = 1 to 75."}]},{"type":"text","order":4,"title":"State final answer","content":"The graphing calculator outputs the final sum:\n\n$$S = 11\\,625$$\n\nThis confirms the analytical calculation of the arithmetic series sum."}]}],"generatedAt":"2026-06-09T08:01:10.012Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"9ae90cf4-e801-4c54-90f0-26be65141b25","revisionId":"eb3bb87e-83d0-44b4-9a22-395ccd69c69a","questionNumber":null,"questionType":"LA","specification":"A series of robotic pulses is programmed such that the energy, \$ E_k \$, of the \$ k \$-th pulse follows a geometric sequence with first term \$ a \$ Joules and common ratio \$ r \$, where \$ r > 0 \$.\n\nGiven that\n- the 2nd pulse has energy \$ 108 \$ Joules\n- the 4th pulse has energy \$ 48 \$ Joules","difficulty":4,"difficultyBand":"medium","tier":null,"totalMarks":7,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"6cfec091-2f66-4aca-9705-769237196435","label":"a","content":"Show that \$ r = \\frac{2}{3} \$.","markscheme":"- Uses $E_2 = ar = 108$ and $E_4 = ar^3 = 48$ to write $\\frac{ar^3}{ar} = \\frac{48}{108}$ M1\n- Simplifies to $r^2 = \\frac{4}{9}$ and concludes $r = \\frac{2}{3}$ since $r > 0$ A1","marks":2,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"f8005a32-51f7-4762-8a45-33ccb4217f5a","label":"b","content":"Determine the value of \$ a \$.","markscheme":"- Substitutes $r = \\frac{2}{3}$ into $ar = 108$ to find $a = 162$ B1","marks":1,"order":1,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"816d73c3-9983-4c01-b039-5fe860d16e24","label":"c","content":"Given that the total energy released by the first \$ n \$ pulses is greater than \$ 480 \$ Joules,\n\nfind the smallest possible value of \$ n \$.\n\n(Solutions based entirely on graphical or numerical methods are not acceptable.)","markscheme":"- Substitutes $a = 162$ and $r = \\frac{2}{3}$ into the sum formula $S_n = \\frac{a(1 - r^n)}{1 - r}$ to set up the inequality $\\frac{162(1 - (2/3)^n)}{1 - 2/3} > 480$ M1\n- Simplifies to $486(1 - (2/3)^n) > 480$, leading to $(2/3)^n < \\frac{6}{486}$ or $(2/3)^n < \\frac{1}{81}$ A1\n- Applies logarithms to solve for $n$, e.g. $n \\ln(2/3) < \\ln(1/81)$ or $n > \\frac{\\ln(1/81)}{\\ln(2/3)}$ M1\n- Obtains $n > 10.838...$ and concludes the smallest integer value is $n = 11$ A1","marks":4,"order":2,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"a_level_maths__wjec","summaries":[{"id":"0429ea80-c16f-442c-a8c7-9db5d90fae0d","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"44c49e73-5e0e-4d0c-9ae6-028714ab7efe","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"1316c378-f476-466e-8fc5-f00c518af57c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"15882735-8366-494e-b31e-147436db8d66","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"1c79b4d0-a538-4bd9-9727-e36a02c34b4e","difficultyBand":"easy","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"2dbce232-c0d7-4dd6-b139-3eeb4b6a2e55","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"3950bc86-f600-4c92-a793-037add492875","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"46020c28-8d82-472a-be22-1498e11265cb","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"4af758f8-ac50-4194-a926-01b49e3830c4","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"567a339e-ed5f-49fa-b1f0-ea25bac54256","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"5b95689d-71eb-476e-a8e6-a38dbb8ea544","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"5d65fca2-3aa5-4015-9b4d-9e3550438b7c","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"5ef3379b-770b-4c55-9ce0-1e620c699887","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"5fb7e131-2dc1-42ac-b9ce-d47ab4f6613e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"7d5b1bae-eaa4-43a7-9fa6-2acd10701082","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"842a134f-0c6b-4eca-b37d-70f1a75095c1","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"926eb417-4bb8-4613-9fba-78eb249f72a7","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"a040b0b7-7843-47a7-b7ee-725bb314c8c1","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"a76fc420-fda5-4857-adf5-ea2913be24c2","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"af8190d3-2895-4d92-a1f1-cd21e0877050","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b1721fba-4e52-4d3f-ade1-a7f38c98a371","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"c1feb070-8b01-4aab-a059-0b23dc16bc0f","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c408e432-3b7c-49ca-b448-5bb7f33922c6","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"cd105ce9-8aca-4564-af12-6527215f8d75","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"d2616403-974a-4433-8a7d-26f3adfdedb8","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"e51738ee-5f60-4d42-a266-711422a5f499","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"efd5539a-7834-498d-90fa-afdf1d655594","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"fbe1dffe-6f64-43ed-83ec-dacad2c25461","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"fddc2bfd-83f8-4ddb-8167-514e0cb2afdd","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"ff1cf253-168e-45cd-a8ff-6650870301e7","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"d4d32f37-b31e-4754-8a1e-d4ec4eb9e593","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"e9e3f1f0-d8d1-4186-b489-9874e4a80123","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"02e9df4b-1c3b-4086-89c5-ff1e95c0f4d4","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"039ad70c-5bfd-43d0-9b1e-b3edcc8e6a18","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"0a2e7cbe-616f-4e28-8cad-89dedc53b091","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"364a708c-629e-44a6-8918-aaec1fc53d26","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"3c9a5665-b0bc-44a9-9d81-524d098b4aff","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"439f90b7-0ba8-4073-bb22-26cf11758bc7","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"43adacca-2515-4d67-9e2d-ab3f5f930df2","difficultyBand":"easy","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"5036cd17-4f44-42fa-96cf-4039a69ac893","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"584871c6-110d-429e-8e67-ba48f1b8a028","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"5856c46d-c26f-4278-8ec6-dbdc966858f4","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6be3b15e-8805-4145-87e5-40193e6921e2","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"6e49b94b-406d-48b2-b99b-d7b4d2e5e018","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"9a64a5e8-5b50-4c61-b1a7-eaff53ef94bc","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"9dce6344-a0ec-4c3e-b678-bc05276042ea","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"a2e05815-84d8-4df1-b228-38d7561163b2","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"a5b5e98b-08ad-46db-add6-67dcc7c9a7f7","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b89ef1bd-dda8-4f47-b1dc-850ee3d100b3","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"bd796b60-06f3-4c8b-b19a-de1132ff315e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"be322240-35e8-48a6-9009-32e7d67c117a","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"c2de01f5-7009-4ab1-8043-02e8044e34aa","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"cf9de885-9548-451d-b5fc-0d133c94c5d3","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"d1844edc-4a78-4bae-86e8-b2ed9e8295b9","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"d2aa5ca0-f062-4901-a89c-2afcec57d15e","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"eb9eaff0-b538-4f88-99de-ae22ee7a57e7","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f2722076-a86b-4b54-8669-6465ee6dcaef","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f3a77a34-359e-475f-a167-75d0708d5d23","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"fa58538e-223c-45f9-acfa-0e88eb71ec83","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"fc4dbba4-804d-41f1-bf07-dba16312c268","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"15e28dd3-5249-45ae-937f-e6d30570dfb7","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"36ff5ffd-5e92-4a14-acbf-aa56d5684f84","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"04f8db56-a5fa-451d-b5da-e9de16070569","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0fc08c15-4d31-42fa-9604-298af3551c6d","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"23334570-6c27-4879-87e0-3cd60360b284","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"27b1e794-cd22-46a8-bace-759ab0eff649","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"2dbb9750-3eca-43ea-ad7f-8afb4ae70e9b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"32da88da-2fe8-4915-92dc-63e597e40d85","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"3e055061-fab9-4b24-9e84-4d7e0da8032c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"47448884-1ce2-404a-83a8-eb98ca7a0450","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"5176cffa-ebf6-4b7a-82ea-40993e88a60c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"5948e658-ca9a-4843-8f6c-818f0212f8fa","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"5c43ec0e-a03b-449f-a2c0-301ca0532efc","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"60bb4244-2f43-44be-895f-66a645d8d9fd","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"626e5417-01b1-4503-83ee-86627c92915f","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"67ca04dc-da0f-4cad-9b51-1f7dd073087b","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"691fcfa7-8149-4d19-977d-1df2501252ac","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"6a97464c-423a-4d98-bb39-32c2662eff10","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"770290bc-6bd9-42b5-8061-213063218b09","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"99447d6d-aed8-474f-b6d1-01a7abc0d1b0","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"aff5e867-ea3b-4a26-9a7c-0af68ae1a615","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"b1c78b39-7ff3-4d13-a0c8-6383e71a97bd","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"b3e4d574-6fab-4d21-bd8c-508e753f2e44","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d15077cd-0ffd-4b71-ad46-aa0bbb29314d","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d6323517-5695-4fb6-8567-391d812944e3","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"d82b8787-61cd-4db3-a84e-42c9e0c6d7c5","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"ef9f3562-22be-429e-a8e5-272e03765a58","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"eff92c51-143d-42c5-b251-84d5e942c301","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"fc34957f-5d74-4458-8dec-afa003423b8a","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7f1b3d00-0054-49da-9554-dda44ff2e4eb","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"a8f1f156-5041-417a-a02d-83f7b9b5e24c","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"660356b0-abd6-4b06-b3f5-32c9c953e8db","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"8ed2c2a7-689d-4be9-9bf1-d783cb4c35ec","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"921d4878-f4dc-4f36-8e5a-813cf59d7070","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"80eabdc6-7592-4ea8-8523-9a1de827dad7","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"34c14ad2-c7d5-4a7a-a6a7-c40f83d80287","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"0243fffd-1f86-401e-98f7-2a1b40d3a95b","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"03311c28-d437-405e-aaa2-fe9ef6ba38da","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"040d3700-91b5-455a-9d23-6e5a0eb1b339","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"076792c8-14d4-495d-ac2a-629067b2aecf","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"085f9a68-e028-4455-9161-bcfe40868b86","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"09813320-60bb-40f1-99d7-b9748ca90a2f","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"0bd25f07-2b38-4f79-95c0-8c14e78dff1e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0be20644-fa0d-4621-9eca-55e217454648","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0cca90c8-7ed5-4bdd-828e-598fa8d1f577","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"0edf0c84-0948-4b4c-bfa7-84f2409c5094","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"10d11d25-b3b6-44e1-8022-8b11904efeeb","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"1679e6ed-4e3c-4a22-832f-ba18c986a3fb","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"16d4a8ff-b8d0-4942-89cb-29f4c8cbd823","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"17ff9f32-b107-4645-97c3-ae6e34f27f26","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"1a973505-c38b-41cb-8071-5dbf532bf708","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"1c23cdc8-663b-4c41-be3a-33c946a340ba","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"1e4190ca-ae40-4abb-84b5-ea63dbf0369b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"2494deb2-5bb4-4ac1-b79e-59712e7ccc13","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"25fc68d0-c943-46b8-b17c-2d726a27fd95","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"269a3aca-85e7-4b04-99dc-53f68f5971bf","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"2d180c3d-5e91-4921-b7d0-df95ec1d9dcd","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"2e6bfee7-d832-4871-992e-3f03f37d8da2","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"34e75bf8-7fca-4a81-ac12-79c8b2ed4a49","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"3a29f7e7-f3bb-40fa-b7b8-061959400b98","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":false},{"id":"3bb384de-ae9c-47d9-8fb9-7151c9fe8d74","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3c05112e-6358-4470-a53f-1e101c2a2266","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"3e2ea611-b792-468b-813c-ed0846432da8","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"3ebd1eb5-abb1-4fcc-bda4-692b4a784966","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"4139864f-c551-437f-be46-9222f2533947","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"414a7b99-1f37-4b28-b01c-52b671747e8d","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"4150465f-f505-4bb8-b256-cf4ef043ec9b","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"41fd9cfe-2990-4645-bf51-56d0a068e4fe","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"4385428a-c8d1-4c62-a40a-6fe26604b610","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"43e9da4a-f838-44c9-a275-b889d93f1bd3","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"47e6f8b7-afd2-4d49-9a68-87b49a47134c","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"48f7f263-6725-4a24-9b12-348bae5c114c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"4bbfc898-994a-4d1e-ba76-86c5c64ebe12","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"4cf822f8-e52c-4be8-9245-b4e2eba800ef","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"4e52a92e-73be-4fae-8521-7b885a5de797","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"50fb24cd-b977-4df9-a490-f64225fac42e","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"517d0c19-d597-4110-864a-489a36185cd4","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":false},{"id":"5494b0d2-f288-43ce-bc79-49a31a59b889","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"5547d80d-d7ff-4af3-a328-7e1e59c1725b","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"561b816b-fd55-4fc7-b9cd-02dd65433c8d","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"56c22dba-bf9c-42b3-aba6-66202eb3ffdf","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"56cc26cc-bf5e-4c66-885c-4d375a924175","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"574e4716-037a-4811-a3dd-1a1f206266a9","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"57895e99-b75b-42cf-bb83-4c96acaf4f55","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"58ab2398-682f-4b01-a663-d719899ca3c9","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"5b6aade5-dc87-4be0-a6b6-53406cdbc334","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"5ea5f4ec-6760-4ab6-b3b3-671a5bf4f498","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6174aaa8-8897-4e9e-883d-a49fb3bd7c3b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"64564963-1522-4e6b-824d-d8e9ee1dbdaf","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"65c845f0-0cc7-4992-86c3-f4d490f88993","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"66bed176-cf14-455f-8600-e4499e9c3e21","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"6702cb6f-d7a0-4203-b898-4051aa7a809c","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"6c450847-aec7-489c-9329-283c1351e741","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"6c69b838-a08c-4a95-9c34-b92923f2ba4d","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"6d9cdf3d-2465-452b-8aa3-ee4dd81a0b24","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6f4b7d94-6d8c-42c3-82d5-3f539574de55","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"70342bba-bbd5-47d0-909f-5bbe9aedd386","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"707da3b5-6b4d-43ed-bdd4-8bbd4154d8f9","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"71245a0e-b110-4dbd-8e66-b519f78d5010","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"7b29e5ab-30ed-44ca-8b8f-80a7f8efa445","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"7c4eb826-882e-4c6a-ae10-e73301347088","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"7c5acfc1-2e08-475c-8600-0d9b740ee6c8","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7c7e32e3-3f69-476d-bfc8-4c197aa4cb75","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7eab035a-6de2-42ea-af37-26e10de04f13","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"8067a0dd-9584-4a32-bea8-c352ac90aad8","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"809a29c4-5699-4b7d-bada-1bbfd4b02139","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":false},{"id":"83c01141-37ae-4c9d-a2d3-e8e7b6d6b2f9","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"8424f39d-3fd8-463e-8723-0c7c71539bc7","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"84795db1-9c45-46ca-9f6f-5b3b2e5112ee","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"853b55fe-5893-44eb-8da0-8f4fc8cde649","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"85eb121d-76ce-4454-a713-abbf207d162a","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"867d85bf-3975-4cd5-b9bd-9834f1cbfa08","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"87250aa7-33dc-409b-8d66-c500f7462281","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"873632bf-18b4-4b35-9270-e7a272bd2477","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"87542ee1-189d-4a92-9e71-6ee6c3f2c6f2","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"88505649-fe3a-4a95-9ba0-9e4e7f97e3a3","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"8cb93779-db47-4997-81af-0fffb89dce10","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"91406182-f6e1-4b18-aa42-7302418ed6b3","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"927012b8-7525-4cfd-8ff8-baa4bab09978","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"93e0563f-f1c6-4dbd-b8fa-4da331fd85f4","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"93fd17ae-5ebb-4d63-a1d4-a2c9231875c3","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"96caaf46-ef14-4263-a873-9218415ea6e8","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"98d41dcb-08fe-42f2-aa9f-92f939ea317e","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"9aadfd8a-f1df-4d17-a73d-c82be4a32f66","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9ae90cf4-e801-4c54-90f0-26be65141b25","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"9b166de8-ab8a-4560-a431-d32a43970c20","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"9c955d55-d2f0-4b14-a0df-32dd54237c04","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"a1a63678-e667-4d9a-b477-f28d35a404dd","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"a260d849-af1c-4808-b85f-e91ad21b3a5b","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":true},{"id":"a9ae94e5-fa6f-4114-8221-6a62c840063c","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"aa59e030-c4ba-4386-9b05-a4ed52743c47","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"ab809f58-f25c-4772-b4d0-a25e6271d314","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":true},{"id":"ae0a2a58-db56-4959-b5ed-333bd4f9ae49","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"b0831071-ebb3-44c1-b79c-4649ec914fb9","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c1898193-78b8-46ad-b296-4f85eb483375","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"c1db6651-7722-4a4c-baee-a34d8298b8d5","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"c509aaed-9aeb-4dcd-90a6-972a2f4d7679","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"c5160e74-6e5c-4d5f-8399-60a97f2a00d0","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"c8b60205-1a9a-4c0f-966e-37f30d4f3047","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"c9ca9a99-1338-4b6f-a418-aaa37f022fcc","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"cd0b7b16-939b-4c6e-809c-a965787dd235","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"d3974a89-ab32-442d-ab2e-d3443bcc529d","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"dcb6f220-a63d-4134-b6a6-cdfd15c5382e","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"dcda3916-511f-4f2d-827e-0ccb309c5c69","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"dd328e26-a611-45cc-a6e6-fd760953d17c","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"dd9ef33d-9cde-4654-b32a-de4dde366b77","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"dddbd33e-d3a0-414a-99f1-bf0514ffbbc9","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"de1331d1-ca2b-48d8-8eac-df6477c5fb69","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"dfcc52f8-64cf-4484-bc15-da2e5d8cf459","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e1f03d5c-1de7-4cee-a05c-e1f1c72be328","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"e29aff3f-0f1f-4123-be54-a5b6463df217","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"e34eed40-913f-4368-a36e-9315ab5fdb19","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e39a7e12-bfa5-40ff-bdfc-6111462d56b8","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":true},{"id":"e44f63e4-2574-4b4b-90cd-9c821e6758f1","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e4dc4e6e-e73f-4f2a-9b84-2a3f64e17bff","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"e9567463-99f6-4eea-9693-f9d191dc3fbb","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"e9f373a6-edc7-4482-8124-b02a09ace134","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"ea741d59-dafb-496a-af29-6435f92f4b31","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"edc93ae5-1408-4dfe-80a0-3ae93fc3495c","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"ef7ce60d-b1e5-499b-9737-9b955ef30cd2","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"f11eb173-a8e4-4301-9a9d-9a703ff9f770","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"f2ba430c-e0d4-4a61-b869-ebfba011248a","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"f6708626-e41a-4e78-92c2-d88deb197016","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"f6ea2199-25f6-4cfc-8a61-e6e9b9d386a8","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":false},{"id":"f6fbe3c9-9cb6-457c-ac06-186ff51e3da2","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"f96adf29-ce4a-495b-a8e4-86447332f02b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"fafc9b4d-c889-4db4-843e-0b132b404089","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"fb5a3e08-6212-4abb-89c3-747ee401e88b","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"fbc0c68f-6b93-4ad1-a738-7c648a2a1535","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"fd12101c-cc78-4fe6-933e-f491d34aeb40","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"ff4703ce-b55d-4c2f-bfe7-df2271eddc4d","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true}],"title":"3.4 Sequences and Series (A-level only)","topicId":"7e0865d9-37e1-4f64-a1e9-cc8d221ebcdb","topicSidebar":"$L3a","topicSlug":"sequences-and-series","topicTitle":"3.4 Sequences and Series (A-level only)"}]}],"$L3b","$L3c"]}]

3.4 Sequences and Series (A-level only)

Question 144

3.4 Sequences and Series (A-level only) Questions

3.4 Sequences and Series (A-level only)

Question 144

3.4 Sequences and Series (A-level only) Questions