Proof by induction: past paper questions
Real questions on proof by induction
The newest 20 of 46 written questions. Open one to see it on the real paper.
- 9231/12 · October/November 2025 · Q3a[5]
5-mark question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/12 · October/November 2025 · Q3b[3]
3-mark “Show” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/13 · October/November 2025 · Q3[7]
7-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/11 · May/June 2025 · Q3a[5]
5-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/11 · May/June 2025 · Q3b[2]
2-mark “Deduce” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/12 · May/June 2025 · Q3a[5]
5-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/12 · May/June 2025 · Q3b[2]
2-mark “Deduce” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/13 · May/June 2025 · Q2[6]
6-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/11 · October/November 2024 · Q2[6]
6-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/12 · October/November 2024 · Q1[5]
5-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/13 · October/November 2024 · Q2[6]
6-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/11 · May/June 2024 · Q2[6]
6-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/12 · May/June 2024 · Q2[6]
6-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/13 · May/June 2024 · Q4a[5]
5-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/13 · May/June 2024 · Q4b[6]
6-mark “Show” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/11 · October/November 2023 · Q2[6]
6-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/11 · May/June 2023 · Q1a[5]
5-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/12 · May/June 2023 · Q1a[5]
5-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/13 · May/June 2023 · Q1[6]
6-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper → - 9231/11 · October/November 2022 · Q2[6]
6-mark “Prove” question — its maths and diagrams are best seen on the paper.
See it on the paper →
What examiners said
The majority of candidates understand the structure of an induction proof. To demonstrate the base case, a link to the statement in the case n =1 needed to be shown. The hypothesis was often written correctly, and many remembered to state that this is an assumption. Those who worked with 2A k +1 = 2 A k .A or 2 A k +1 = A.2 A k managed the matrix multiplication correctly, mostly showing an interm...
— Examiner report, May/June 2023, 9231/12 Q1(a) (Cambridge)
The majority of candidates understand the structure of an induction proof and the base case was usually proved correctly. The wording of the hypothesis was often incomplete. Candidates need to assume that 53 k + 32k − 33 is divisible by 31 for some positive integer k, rather than just assume that 53 k + 32k − 33 is true. The candidates who attempted to show that the (k + 1)th term is itself divi...
— Examiner report, May/June 2023, 9231/13 Q1 (Cambridge)
The structure of a proof by induction was clearly well understood and many solutions earned high marks. Those candidates who started with the expression for the kth term being divisible by 48 and moved towards the expression for the (k + 1)th term being divisible, were usually correct in their algebra. Those who showed that the difference between the two terms is divisible by 48 needed to remember...
— Examiner report, October/November 2022, 9231/11 Q2 (Cambridge)
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