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Every odd number can be written as $2n+1$, so use $2n+1$ for the first odd number. Stating that $n$ is an integer matters because this is a proof for all consecutive odd numbers.","highlights":[{"text":"3 consecutive odd numbers","location":"specification"}]},{"type":"text","order":1,"title":"Write consecutive terms","content":"Consecutive odd numbers differ by $2$. Therefore, the three numbers are $2n+1$, $2n+3$ and $2n+5$. Their sum is\n\n$$(2n+1)+(2n+3)+(2n+5).$$\n\nWriting this expression earns **M1**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=sum_expression}{S=(2n+1)+(2n+3)+(2n+5)\\qquad}}$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=simplified_sum}{S=6n+9\\qquad}}$"},{"latex":"$$\\color{@orange}{\\htmlData{anchor=factorised_sum}{S=3(2n+3)\\qquad}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sum_expression"},{"color":"sky","style":"text-color","anchor":"simplified_sum"},{"color":"orange","style":"text-color","anchor":"factorised_sum"}],"connections":[{"to":"simplified_sum","from":"sum_expression","color":"sky","label":"collect terms"},{"to":"factorised_sum","from":"simplified_sum","color":"orange","label":"factorise"}]},"order":2,"title":"Simplify and factorise","description":"Collect the three n-terms and the constants, then take out the common factor 3. Simplifying to 6n + 9 and factorising to 3(2n + 3) earns A1."},{"type":"text","order":3,"title":"State the conclusion","content":"Because $2n+3$ is an integer, $3(2n+3)$ is divisible by $3$. This proves that the sum is always a multiple of $3$."}]}],"generatedAt":"2026-07-20T16:51:40.534Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"60776e04-49d2-4f4f-b0b5-b6f42c10c886","revisionId":"5428a55e-d1da-4906-a23f-6526ce39c841","questionNumber":"8","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":2,"sourceTitle":"Proof","calculatorUse":null,"isCalculation":true,"parts":[{"id":"f160b8e7-41ae-49c1-aecc-cc736252dc47","label":"0","content":"Prove that the sum of 3 consecutive even numbers is always a multiple of 6.","markscheme":"- Expression for sum: $2n + (2n + 2) + (2n + 4)$ M1\n- Simplifies to $6n + 6$ and factors to $6(n + 1)$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Writes an algebraic expression for the sum of three consecutive even numbers.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies to $6n + 6$ and factors out 6.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-07T02:12:13.876Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"ed073231-38de-4ff8-a243-3a7354cbaa77","revisionId":"f0dc0e4f-a781-4838-b3a3-cfd15d273245","questionNumber":"9","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":2,"sourceTitle":"Proof","calculatorUse":null,"isCalculation":true,"parts":[{"id":"c0993e7d-2638-4b6b-99c6-0cd1657571f4","label":"0","content":"Prove algebraically that the sum of the squares of any 2 even positive integers is always a multiple of 4.","markscheme":"- Expression for sum of squares: $(2n)^2 + (2m)^2$ M1\n- Simplifies to $4n^2 + 4m^2$ and factors to $4(n^2 + m^2)$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Writes an algebraic expression for the sum of the squares of two even integers using different variables.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies to $4n^2 + 4m^2$ and factors out 4.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-07T02:12:13.876Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths__oxford_aqa","summaries":[{"id":"053eadb7-afee-4fdc-a3ba-b1adb85ca8af","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"c228f89f-55c0-4d4f-85fa-b3feacf5899d","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"4399b87b-8eb0-4be3-8ec2-c2c48287b28b","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"6a6708d6-01a0-4c21-91a6-26d31a9a6a4f","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"86f81a91-c3fb-409f-ac98-7f360b23fb33","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"7e6dc2b0-8be4-4907-b1ed-cf321d8b5c0a","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"45903ecf-5d89-48e3-9fd1-fe8573654698","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"60776e04-49d2-4f4f-b0b5-b6f42c10c886","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"ed073231-38de-4ff8-a243-3a7354cbaa77","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"0b983462-5590-4395-9ed5-1ef9ec2af03e","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"0e1344db-9d9a-4049-8613-03673b5bbe76","difficultyBand":"hard","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1c00cceb-6a5d-42ff-9222-8367ed56b68e","difficultyBand":"hard","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"5764d86e-2595-4647-a0d7-c04c6780066a","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3d3f86db-d4c2-4011-80d6-4780cfc9ce2a","difficultyBand":"hard","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true}],"title":"Proof","topicId":"dc605650-9188-4db2-815c-e346a991928a","topicSidebar":"$L3a","topicSlug":"grade-8-9-8-3-proof","topicTitle":"Proof"}]}],"$L3b","$L3c"]}]