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So start with\n\n$$(n+2)(n+1)+(n+2).$$\n\nTo prove the claim for every integer $n$, keep $n$ algebraic rather than testing particular values.","highlights":[{"text":"$$n$ is an integer.","location":"specification"},{"text":"the sum of $(n + 2)(n + 1)$ and $n + 2$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=whole_start}{(n+2)(n+1)+(n+2)}}$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=whole_expanded}{n^2+3n+2+n+2}}$"},{"latex":"$$\\color{@amber}{\\htmlData{anchor=whole_collected}{n^2+4n+4}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"whole_start"},{"color":"sky","style":"text-color","anchor":"whole_expanded"},{"color":"amber","style":"text-color-with-background","anchor":"whole_collected"}],"connections":[{"to":"whole_expanded","from":"whole_start","color":"sky","label":"expand the product"},{"to":"whole_collected","from":"whole_expanded","color":"amber","label":"collect like terms"}]},"order":1,"title":"Expand and simplify","description":"Expand only the product; the separate term is added unchanged. Then collect the like terms. Reaching the simplified quadratic earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@green}{\\htmlData{anchor=identity}{a^2+2ab+b^2=(a+b)^2}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=application}{n^2+4n+4=(n+2)^2}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"identity"},{"color":"green","style":"text-color-with-background","anchor":"application"}],"connections":[{"to":"application","from":"identity","color":"green","label":"take a=n and b=2"}]},"order":2,"title":"Factorise the quadratic","description":"Use the perfect-square identity, taking a to be n and b to be 2. Writing the result as a square earns A1."},{"type":"text","order":3,"title":"Complete the proof","content":"Because $n$ is an integer, $n+2$ is also an integer. Therefore $(n+2)^2$ is the square of an integer, so the original sum is **always a square number**, as required.","highlights":[{"text":"always a square number.","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:42:54.840Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"45903ecf-5d89-48e3-9fd1-fe8573654698","revisionId":"1817a804-6345-4729-a212-1e5a87fd9fca","questionNumber":"7","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":2,"sourceTitle":"Proof","calculatorUse":null,"isCalculation":true,"parts":[{"id":"172105bd-9d91-46ce-83a8-9003877457a6","label":"0","content":"Prove that the sum of 3 consecutive odd numbers is always a multiple of 3.","markscheme":"- Expression for sum: $(2n + 1) + (2n + 3) + (2n + 5)$ M1\n- Simplifies to $6n + 9$ and factors to $3(2n + 3)$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Writes an algebraic expression for the sum of three consecutive odd numbers.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies to $6n + 9$ and factors out 3.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-07T02:12:13.876Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"algebraic_proof","label":"Algebraic proof","steps":[{"type":"text","order":0,"title":"Represent the first number","content":"Let $n$ be an integer. 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}],"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"]}]