Proof by induction: past paper questions

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• Question 2May/June 2026, paper 11Structured6 marks

  Prove by mathematical induction that $3^{6n}+{92}^n+89$ is divisible by 91 for all positive integers $n$.

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• Question 2May/June 2026, paper 12Structured6 marks

  Prove by mathematical induction that $3^{6n}+{92}^n+89$ is divisible by 91 for all positive integers $n$.

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• Question 2Oct/Nov 2023, paper 12Structured6 marks

  Prove by mathematical induction that, for all positive integers n, (dⁿ)/(dxⁿ)(x² e^x) = (x² + 2nx + n(n-1))e^x.

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• Question 2Oct/Nov 2024, paper 13Structured6 marks

  Prove by mathematical induction that, for all positive integers $n$, $$\frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}({\tan }^{-1}x)=P_n(x)(1+x^2{)}^{-n},$$ where $P_n(x)$ is a polynomial of degree $n-1$.

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• Question 2May/June 2024, paper 11Structured6 marks

  Prove by mathematical induction that $6^{4n}+{38}^n-2$ is divisible by 74 for all positive integers $n$.

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• Question 2May/June 2025, paper 13Structured6 marks

  Prove by mathematical induction that ${2025}^n+{47}^n-2$ is divisible by 46 for all positive integers $n$.

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• Question 1May/June 2023, paper 13Structured6 marks

  Prove by mathematical induction that, for all positive integers $n$, $5^{3n}+{32}^n-33$ is divisible by 31.

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