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(Higher)","calculatorUse":null,"isCalculation":true,"parts":[{"id":"0e580a94-f666-4f1b-99c5-0c96ae32768a","label":"","content":"The sum of the first 48 terms of an arithmetic series is 4 times the sum of the first 36 terms of the same series. \nFind the sum of the first 30 terms of this series.","markscheme":"- Sets up the equation $S_{48} = 4(S_{36})$ using the sum formula M1\n- Expands and simplifies: $24(2a + 47d) = 72(2a + 35d)$ M1\n- Simplifies to find a relationship between $a$ and $d$, e.g., $2a + 29d = 0$ M1\n- Recognises that $S_{30} = 15(2a + 29d)$ M1\n- Final answer 0 A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Correctly applies the sum formula to both $S_{48}$ and $S_{36}$.","code":"M"},{"id":"p2","marks":1,"description":"Equates $S_{48}$ to $4 \\times S_{36}$ and begins simplification.","code":"M"},{"id":"p3","marks":1,"description":"Simplifies the equation to reach $2a + 29d = 0$ or equivalent.","code":"M"},{"id":"p4","marks":1,"description":"Identifies that the expression for $S_{30}$ contains the factor $(2a + 29d)$.","code":"M"},{"id":"p5","marks":1,"description":"Final answer of 0.","code":"A"}],"finalAnswers":["0"],"generatedAt":"2026-05-07T02:11:27.220Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"arithmetic_series_formula","label":"Use the sum formula","steps":[{"type":"text","order":0,"title":"Recall the required formula","content":"This question uses the arithmetic-series sum formula, which is required prior knowledge before this method:\n\n$$S_n=\\frac{n}{2}\\bigl(2a+(n-1)d\\bigr).$$\n\nHere, $a$ is the first term and $d$ is the common difference (the amount added each time).","highlights":[{"text":"The sum of the first 48 terms of an arithmetic series is 4 times the sum of the first 36 terms of the same series.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=setup}{\\color{@sky}{S_{48}=4S_{36}}}$"},{"latex":"$$\\htmlData{anchor=substitution}{\\color{@amber}{\\frac{48}{2}(2a+47d)=4\\left[\\frac{36}{2}(2a+35d)\\right]}}$"},{"latex":"$$\\htmlData{anchor=simplified}{\\color{@green}{24(2a+47d)=72(2a+35d)}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"setup"},{"color":"amber","style":"text-color","anchor":"substitution"},{"color":"green","style":"text-color-with-background","anchor":"simplified"}],"connections":[{"to":"substitution","from":"setup","color":"sky","label":"apply the sum formula","style":"solid"},{"to":"simplified","from":"substitution","color":"orange","label":"simplify coefficients","style":"solid"}]},"order":1,"title":"Substitute into the formula","description":"Set the two sums in the required ratio, then apply the arithmetic-series sum formula. Writing S₄₈ = 4S₃₆ using the sum formula earns the first M1; simplifying to the final line earns the second M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=expanded}{\\color{@sky}{48a+1128d=144a+2520d}}$"},{"latex":"$$\\htmlData{anchor=collected}{\\color{@red}{-96a-1392d=0}}$"},{"latex":"$$\\htmlData{anchor=relationship}{\\color{@green}{2a+29d=0}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"expanded"},{"color":"red","style":"text-color-with-background","anchor":"collected"},{"color":"green","style":"text-color-with-background","anchor":"relationship"}],"connections":[{"to":"collected","from":"expanded","color":"red","label":"collect terms","style":"solid"},{"to":"relationship","from":"collected","color":"green","label":"divide by negative 48","style":"solid"}]},"order":2,"title":"Find the key relationship","description":"Expand both brackets and collect terms on one side. Dividing by negative 48 gives the relationship needed for the 30-term sum. Deriving 2a + 29d = 0 earns the third M1."},{"type":"text","order":3,"title":"Set up the required sum","content":"Now use the same formula with $n=30$:\n\n$$S_{30}=\\frac{30}{2}\\bigl(2a+(30-1)d\\bigr)=15(2a+29d).$$\n\nRecognising this form earns the fourth **M1**. It contains exactly the relationship found in the previous step.","highlights":[{"text":"Find the sum of the first 30 terms of this series.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=sumform}{\\color{@sky}{S_{30}=15(2a+29d)}}$"},{"latex":"$$\\htmlData{anchor=zeroform}{\\color{@green}{S_{30}=15(0)}}$"},{"latex":"$$\\htmlData{anchor=answer}{\\boxed{\\color{@red}{S_{30}=0}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sumform"},{"color":"green","style":"text-color","anchor":"zeroform"},{"color":"red","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"zeroform","from":"sumform","color":"green","label":"use the relationship","style":"solid"},{"to":"answer","from":"zeroform","color":"sky","label":"evaluate","style":"solid"}]},"order":4,"title":"Use the relationship","description":"Use the relationship already established to evaluate the 30-term sum."}]}],"generatedAt":"2026-07-21T03:59:00.258Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":42},"walkthroughVideo":null}],"subjectCode":"igcse_maths__cie","summaries":[{"id":"325e26e3-a2f7-4c1d-a90f-939a6d307ddd","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"fb470127-3e1e-4d6f-b8e1-aab9975c7b41","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"0900cc55-1d40-42fe-bb15-472d8077ab99","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"48a4915b-ca6c-4d82-aaa9-1da6afab8afc","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true}],"title":"Sequences (Higher)","topicId":"b91eb0c2-aa6d-4a79-9235-04137f8af988","topicSidebar":"$L3b","topicSlug":"grade-8-9-8-7-sequences-higher","topicTitle":"Sequences (Higher)"}]}],"$L3c","$L3d"]}]

Sequences (Higher)

Question 1

Sequences (Higher) Questions

Sequences (Higher)

Question 1

Sequences (Higher) Questions