4.06a Summation formulae: sum of r, r^2, r^3

190 questions

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AQA Further Paper 2 2023 June Q7
3 marks Standard +0.8
Show that $$\sum_{r=11}^{n+1} r^3 = \frac{1}{4}(n^2 + an + b)(n^2 + an + c)$$ where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]
OCR Further Pure Core 2 2024 June Q4
5 marks Challenging +1.2
In this question you must show detailed reasoning. The series \(S\) is defined as being the sum of the squares of all positive odd integers from \(1^2\) to \(779^2\). Determine the value of \(S\). [5]
OCR Further Pure Core 2 Specimen Q1
4 marks Standard +0.3
Find \(\sum_{r=1}^{n}(r+1)(r+5)\). Give your answer in a fully factorised form. [4]
OCR MEI Further Pure Core AS Specimen Q6
5 marks Standard +0.8
  1. Show that, when \(n = 5\), \(\sum_{r=n+1}^{2n} r^2 = 330\). [1]
  2. Find, in terms of \(n\), a fully factorised expression for \(\sum_{r=n+1}^{2n} r^2\). [4]
WJEC Further Unit 1 Specimen Q3
6 marks Challenging +1.2
Find an expression, in terms of \(n\), for the sum of the first \(n\) terms of the series $$1.2.4 + 2.3.5 + 3.4.6 + \ldots + n(n + 1)(n + 3) + \ldots$$ Express your answer as a product of linear factors. [6]
SPS SPS FM Pure 2021 May Q4
3 marks Easy -1.2
Using the formulae for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^2\), show that \(\sum_{r=1}^{10} r(3r - 2) = 1045\). [3]
SPS SPS FM 2021 November Q4
4 marks Moderate -0.3
Prove that $$\sum_{r=1}^{n} 18(r^2 - 4) = n(6n^2 + 9n - 69).$$ [4 marks]
SPS SPS FM Pure 2023 February Q1
4 marks Moderate -0.8
Find \(\sum_{r=1}^{n}(2r^2 - 1)\), expressing your answer in fully factorised form. [4]
SPS SPS FM Pure 2023 November Q6
5 marks Standard +0.8
In this question you must show detailed reasoning. In this question you may assume the results for $$\sum_{r=1}^{n} r^3, \quad \sum_{r=1}^{n} r^2 \quad \text{and} \quad \sum_{r=1}^{n} r.$$ Show that the sum of the cubes of the first \(n\) positive odd numbers is $$n^2(2n^2 - 1).$$ [5]
SPS SPS FM Pure 2024 February Q4
6 marks Challenging +1.2
Using standard summation of series formulae, determine the sum of the first \(n\) terms of the series \((1 \times 2 \times 4) + (2 \times 3 \times 5) + (3 \times 4 \times 6) + \ldots\) where \(n\) is a positive integer. Give your answer in fully factorised form. [6]
SPS SPS FM Pure 2025 February Q11
8 marks Challenging +1.8
The infinite series \(C\) and \(S\) are defined by $$C = \cos \theta + \frac{1}{2}\cos 5\theta + \frac{1}{4}\cos 9\theta + \frac{1}{8}\cos 13\theta + \ldots$$ $$S = \sin \theta + \frac{1}{2}\sin 5\theta + \frac{1}{4}\sin 9\theta + \frac{1}{8}\sin 13\theta + \ldots$$ Given that the series \(C\) and \(S\) are both convergent,
  1. show that $$C + iS = \frac{2e^{i\theta}}{2 - e^{4i\theta}}$$ [4]
  2. Hence show that $$S = \frac{4\sin \theta + 2\sin 3\theta}{5 - 4\cos 4\theta}$$ [4]
OCR Further Pure Core 2 2018 March Q3
3 marks Standard +0.3
In this question you must show detailed reasoning. Use the formula \(\sum_{r=1}^n r^2 = \frac{1}{6}n(n+1)(2n+1)\) to evaluate \(121^2 + 122^2 + 123^2 + \ldots + 300^2\). [3]
Pre-U Pre-U 9795/1 2013 November Q12
10 marks Challenging +1.3
    1. Use the method of differences to prove that $$\sum_{n=k}^N \frac{1}{n(n+1)} = \frac{1}{k} - \frac{1}{N+1}.$$ [4]
    2. Deduce the value of \(\sum_{n=k}^{\infty} \frac{1}{n(n+1)}\) and show that \(\sum_{n=k}^{\infty} \frac{1}{(n+1)^2} < \frac{1}{k}\). [3]
  2. Let \(S = \sum_{n=1}^{\infty} \frac{1}{n^2}\). Show that \(\frac{205}{144} < S < \frac{241}{144}\). [3]
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]
Pre-U Pre-U 9795 Specimen Q4
6 marks Standard +0.3
Write down the sum $$\sum_{n=1}^{2N} n^3$$ in terms of \(N\), and hence find $$1^3 - 2^3 + 3^3 - 4^3 + \ldots - (2N)^3$$ in terms of \(N\), simplifying your answer. [6]