Prove divisibility

A question is this type if and only if it asks to prove by induction that an expression involving powers (e.g., 7²ⁿ - 1, 5ⁿ + 8n + 3) is divisible by a given integer.

68 questions · Standard +0.3

4.01a Mathematical induction: construct proofs
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AQA Further Paper 2 2019 June Q10
7 marks Standard +0.3
Prove by induction that \(f(n) = n^3 + 3n^2 + 8n\) is divisible by 6 for all integers \(n \geq 1\) [7 marks]
AQA Further Paper 2 2023 June Q12
6 marks Standard +0.3
The function \(f\) is defined by $$f(n) = 3^{3n+1} + 2^{3n+4} \quad (n \in \mathbb{Z}^+)$$ Prove by induction that \(f(n)\) is divisible by 19 for \(n \geq 1\) [6 marks]
AQA Further Paper 2 Specimen Q6
5 marks Standard +0.3
Prove that \(8^n - 7n + 6\) is divisible by 7 for all integers \(n \geq 0\) [5 marks]
WJEC Further Unit 1 Specimen Q1
7 marks Standard +0.8
Use mathematical induction to prove that \(4^n + 2\) is divisible by 6 for all positive integers \(n\). [7]
SPS SPS FM Pure 2021 June Q8
6 marks Standard +0.3
Prove by induction that, for \(n \in \mathbb{Z}^+\) $$f(n) = 2^{n+2} + 3^{2n+1}$$ is divisible by 7 [6]
SPS SPS FM 2020 September Q6
5 marks Standard +0.8
A sequence is defined by \(U_n = 2^{n+1} + 9 \times 13^n\) for positive integer values of \(n\). Prove by induction that \(U_n\) is divisible by 11. [5]
SPS SPS ASFM Mechanics 2021 May Q2
6 marks Moderate -0.3
Prove by induction that, for \(n \in \mathbb{Z}^+\), $$f(n) = 2^{2n-1} + 3^{2n-1} \text{ is divisible by 5}.$$ [6]
SPS SPS FM Pure 2021 May Q5
5 marks Moderate -0.3
Prove by induction that, for all positive integers \(n\), \(7^n + 3^{n-1}\) is a multiple of \(4\). [5]
SPS SPS FM Pure 2022 June Q9
6 marks Standard +0.3
Prove by induction that for \(n \in \mathbb{Z}^+\) $$f(n) = 4^{n+1} + 5^{2n-1}$$ is divisible by 21 [6]
SPS SPS FM Pure 2023 February Q5
6 marks Standard +0.3
Prove by induction that for all positive integers \(n\) $$f(n) = 3^{2n+4} - 2^{2n}$$ is divisible by 5 [6]
SPS SPS FM 2024 October Q8
6 marks Standard +0.3
Prove by induction that \(2^{n+1} + 5 \times 9^n\) is divisible by 7 for all integers \(n \geq 1\). [6]
SPS SPS FM 2023 October Q8
5 marks Standard +0.8
Prove that \(2^{3n} - 3^n\) is divisible by 5 for all integers \(n \geq 1\). [5]
SPS SPS FM 2024 October Q8
5 marks Standard +0.3
Prove by induction that \(11 \times 7^n - 13^n - 1\) is divisible by \(3\), for all integers \(n > 0\). [5]
SPS SPS FM Pure 2025 June Q10
5 marks Standard +0.3
Prove by induction that \(f(n) = 2^{4n} + 5^{2n} + 7^n\) is divisible by 3 for all positive integers \(n\). [5]
SPS SPS FM 2026 November Q8
5 marks Moderate -0.3
Prove by induction that \(7 \times 9^n - 15\) is divisible by \(4\), for all integers \(n \geq 0\). [5]
SPS SPS FM Pure 2025 September Q2
5 marks Standard +0.3
Prove by induction that \(11 \times 7^n - 13^n - 1\) is divisible by 3, for all integers \(n \geq 0\). [5]
OCR FP1 AS 2021 June Q4
6 marks Standard +0.3
Prove by induction that \(2^{n+1} + 5 \times 9^n\) is divisible by 7 for all integers \(n \geq 1\). [6]
Pre-U Pre-U 9795 Specimen Q2
6 marks Challenging +1.2
It is given that $$\mathrm{f}(n) = 7^n (6n + 1) - 1.$$ By considering \(\mathrm{f}(n + 1) - \mathrm{f}(n)\), prove by induction that \(\mathrm{f}(n)\) is divisible by 12 for all positive integers \(n\). [6]