Prove summation formula

A question is this type if and only if it asks to prove by induction that a summation equals a given closed-form expression (e.g., ∑r² = n(n+1)(2n+1)/6).

38 questions · Standard +0.3

4.01a Mathematical induction: construct proofs
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Edexcel CP1 2024 June Q6
6 marks Standard +0.3
  1. Prove by induction that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
AQA FP2 2008 January Q5
7 marks Standard +0.8
5 Prove by induction that for all integers \(n \geqslant 1\) $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + 1 \right) ( r ! ) = n ( n + 1 ) !$$
AQA Further Paper 2 2022 June Q5
4 marks Standard +0.3
5 Prove by induction that, for all integers \(n \geq 1\), $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \left\{ \frac { 1 } { 2 } n ( n + 1 ) \right\} ^ { 2 }$$ [4 marks]
Edexcel FP1 Q40
5 marks Moderate -0.3
Prove by induction that, for \(n \in \mathbb{Z}^+\), \(\sum_{r=1}^{n} (2r - 1)^2 = \frac{1}{3} n(2n - 1)(2n + 1)\). [5]
AQA FP2 2011 June Q6
8 marks Standard +0.3
  1. Show that $$(k + 1)(4(k + 1)^2 - 1) = 4k^3 + 12k^2 + 11k + 3$$ [2 marks]
  2. Prove by induction that, for all integers \(n \geqslant 1\), $$1^2 + 3^2 + 5^2 + \ldots + (2n - 1)^2 = \frac{1}{3}n(4n^2 - 1)$$ [6 marks]
AQA Further AS Paper 1 2018 June Q10
8 marks Standard +0.8
  1. Prove by induction that, for all integers \(n \geq 1\), $$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n + 1)^2$$ [4 marks]
  2. Hence show that $$\sum_{r=1}^{2n} r(r - 1)(r + 1) = n(n + 1)(2n - 1)(2n + 1)$$ [4 marks]
WJEC Further Unit 1 2018 June Q2
6 marks Standard +0.3
Prove, by mathematical induction, that \(\sum_{r=1}^{n} r(r+3) = \frac{1}{3}n(n+1)(n+5)\) for all positive integers \(n\). [6]
SPS SPS FM 2020 October Q4
5 marks Standard +0.3
Prove by induction that, for \(n \geq 1\), \(\sum_{r=1}^n r(3r + 1) = n(n + 1)^2\). [5]
SPS SPS FM Pure 2025 February Q3
6 marks Standard +0.8
Prove by mathematical induction that \(\sum_{r=1}^{n} (r \times r!) = (n+1)! - 1\) for all positive integers \(n\). [6]
SPS SPS FM Pure 2025 February Q2
6 marks Standard +0.8
Prove by mathematical induction that \(\sum_{r=1}^{n}(r \times r!) = (n + 1)! - 1\) for all positive integers \(n\). [6]
SPS SPS FM 2025 October Q12
6 marks Standard +0.3
Prove by induction that, for all positive integers \(n\), $$\sum_{r=1}^{n}(2r-1)^2 = \frac{1}{3}n(4n^2-1)$$ [6]
SPS SPS FM Pure 2026 November Q2
6 marks Moderate -0.8
Prove by induction that, for all positive integers \(n\), $$\sum_{r=1}^{n}(2r-1)^2 = \frac{1}{3}n(4n^2-1)$$ [6]
Pre-U Pre-U 9795/1 2018 June Q8
8 marks Challenging +1.2
  1. Write down the values of the constants \(a\) and \(b\) for which \(m^3 = \frac{1}{6}m^3(am^2 + 2) - \frac{1}{12}m^2(bm)\). [1]
  2. Prove by induction that \(\sum_{r=1}^{n} r^5 = \frac{1}{6}n^3(n+1)^3 - \frac{1}{12}n^2(n+1)^2\) for all positive integers \(n\). [7]