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(Higher)","calculatorUse":null,"isCalculation":true,"parts":[{"id":"af545ff9-6ca9-4104-9eea-39b758dc3660","label":"","content":"The 4th term of an arithmetic series is 17 \nThe 10th term of the same arithmetic series is 35 \nFind the sum of the first 50 terms of this arithmetic series.","markscheme":"- Formulates two equations for the terms: $a + 3d = 17$ and $a + 9d = 35$ M1\n- Solves for common difference $d = 3$ M1\n- Solves for first term $a = 8$ M1\n- Substitutes $a$, $d$, and $n=50$ into the sum formula $S_n = \\frac{n}{2}(2a + (n-1)d)$ M1\n- Final answer 4075 A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Sets up simultaneous equations for the 4th and 10th terms using $a + (n-1)d$.","code":"M"},{"id":"p2","marks":1,"description":"Correctly calculates the common difference $d = 3$.","code":"M"},{"id":"p3","marks":1,"description":"Correctly calculates the first term $a = 8$.","code":"M"},{"id":"p4","marks":1,"description":"Correctly substitutes values into the arithmetic series sum formula.","code":"M"},{"id":"p5","marks":1,"description":"Final answer of 4075.","code":"A"}],"finalAnswers":["4075"],"generatedAt":"2026-05-07T02:11:27.220Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"simultaneous_equations","label":"Simultaneous equations","steps":[{"type":"text","order":0,"title":"Model each term","content":"For an arithmetic series, the term formula is $a+(n-1)d$, where $a$ is the first term and $d$ is the common difference. Therefore the given terms produce\n\n$$a+3d=17 \\qquad\\text{and}\\qquad a+9d=35.$$ \n\nFormulating both equations earns **M1**.","highlights":[{"text":"The 4th term of an arithmetic series is 17","location":"specification"},{"text":"The 10th term of the same arithmetic series is 35","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=tenth}{a+9d=35}}$"},{"latex":"$$\\color{@orange}{\\htmlData{anchor=fourth}{a+3d=17}}$"},{"latex":"$$\\htmlData{anchor=subtraction}{(a+9d)-(a+3d)=35-17}$"},{"latex":"$$\\color{@purple}{\\htmlData{anchor=simplified}{6d=18}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=dresult}{d=3}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"tenth"},{"color":"orange","style":"text-color","anchor":"fourth"},{"color":"purple","style":"text-color","anchor":"simplified"},{"color":"green","style":"text-color-with-background","anchor":"dresult"}],"connections":[{"to":"subtraction","from":"fourth","color":"orange","label":"subtract this equation"},{"to":"simplified","from":"subtraction","color":"purple","label":"simplify"},{"to":"dresult","from":"simplified","color":"green","label":"divide by 6"}]},"order":1,"title":"Find the difference","description":"Subtract the fourth-term equation from the tenth-term equation. Solving these simultaneous equations gives the common difference, earning M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$a+3\\color{@sky}{\\htmlData{anchor=dvalue}{d}}=17$"},{"latex":"$$a+3(\\color{@sky}{\\htmlData{anchor=inserted}{3}})=17$"},{"latex":"$$\\color{@orange}{\\htmlData{anchor=simplified}{a+9=17}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=aresult}{a=8}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"dvalue"},{"color":"sky","style":"text-color-with-background","anchor":"inserted"},{"color":"orange","style":"text-color","anchor":"simplified"},{"color":"green","style":"text-color-with-background","anchor":"aresult"}],"connections":[{"to":"aresult","from":"simplified","color":"green","label":"subtract 9"}]},"order":3,"title":"Find the first term","description":"Substitute d = 3 into the fourth-term equation, then solve for a. This gives a = 8, earning M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$S_n=\\dfrac{n}{2}\\left(2a+(n-1)d\\right)$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=substitution}{S_{50}=\\dfrac{50}{2}\\left(2(8)+(50-1)(3)\\right)}}$"},{"latex":"$$\\color{@orange}{\\htmlData{anchor=simplified}{S_{50}=25(16+147)}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=answer}{\\boxed{S_{50}=4075}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"substitution"},{"color":"orange","style":"text-color","anchor":"simplified"},{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"simplified","from":"substitution","color":"orange","label":"simplify each part"},{"to":"answer","from":"simplified","color":"green","label":"evaluate"}]},"order":5,"title":"Substitute into the sum","description":"Use the arithmetic-series sum formula with a = 8, d = 3 and n = 50. The correct substitution earns M1, and the correct final value earns A1."}]}],"generatedAt":"2026-07-20T16:30:08.380Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":29},"walkthroughVideo":null},{"id":"48a4915b-ca6c-4d82-aaa9-1da6afab8afc","revisionId":"3eef9e9d-ed46-4025-85fe-2740a8fbdbc8","questionNumber":"4","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":4,"sourceTitle":"Sequences (Higher)","calculatorUse":null,"isCalculation":true,"parts":[{"id":"49ffca7e-4e59-47ce-a40c-c41a16513b81","label":"","content":"$$(2x + 23)$, $(8x + 2)$ and $(20x - 52)$ are three consecutive terms of an arithmetic sequence.\n\nProve that the common difference of the sequence is 12","markscheme":"- Sets up expressions for common difference: $d = (8x + 2) - (2x + 23)$ and $d = (20x - 52) - (8x + 2)$ M1\n- Equates the expressions: $6x - 21 = 12x - 54$ M1\n- Solves for $x$: $6x = 33 \\implies x = 5.5$ M1\n- Substitutes $x$ back to show $d = 6(5.5) - 21 = 12$ A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Creates two expressions for the common difference $d$ in terms of $x$.","code":"M"},{"id":"p2","marks":1,"description":"Equates the two expressions to form a linear equation in $x$.","code":"M"},{"id":"p3","marks":1,"description":"Solves the equation to find $x = 5.5$.","code":"M"},{"id":"p4","marks":1,"description":"Substitutes $x$ back into a difference expression to show $d = 12$.","code":"A"}],"finalAnswers":["12"],"generatedAt":"2026-05-07T02:11:27.220Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":63},"walkthroughVideo":null}],"subjectCode":"igcse_maths__cie","summaries":[{"id":"325e26e3-a2f7-4c1d-a90f-939a6d307ddd","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"fb470127-3e1e-4d6f-b8e1-aab9975c7b41","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"0900cc55-1d40-42fe-bb15-472d8077ab99","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"48a4915b-ca6c-4d82-aaa9-1da6afab8afc","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true}],"title":"Sequences (Higher)","topicId":"b91eb0c2-aa6d-4a79-9235-04137f8af988","topicSidebar":"$L3a","topicSlug":"grade-8-9-8-7-sequences-higher","topicTitle":"Sequences (Higher)"}]}],"$L3b","$L3c"]}]