4.06b Method of differences: telescoping series

262 questions

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OCR FP1 2010 June Q8
9 marks Standard +0.3
  1. Show that \(\frac{1}{\sqrt{r + 2} + \sqrt{r}} = \frac{\sqrt{r + 2} - \sqrt{r}}{2}\). [2]
  2. Hence find an expression, in terms of \(n\), for $$\sum_{r=1}^{n} \frac{1}{\sqrt{r + 2} + \sqrt{r}}.$$ [6]
  3. State, giving a brief reason, whether the series \(\sum_{r=1}^{\infty} \frac{1}{\sqrt{r + 2} + \sqrt{r}}\) converges. [1]
OCR MEI FP1 2006 June Q9
13 marks Standard +0.3
  1. Show that \(r(r+1)(r+2) - (r-1)r(r+1) \equiv 3r(r+1)\). [2]
  2. Hence use the method of differences to find an expression for \(\sum_{r=1}^{n} r(r+1)\). [6]
  3. Show that you can obtain the same expression for \(\sum_{r=1}^{n} r(r+1)\) using the standard formulae for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^2\). [5]
OCR MEI FP1 2007 June Q6
6 marks Moderate -0.8
  1. Show that \(\frac{1}{r+2} - \frac{1}{r+3} = \frac{1}{(r+2)(r+3)}\). [2]
  2. Hence use the method of differences to find \(\frac{1}{3 \times 4} + \frac{1}{4 \times 5} + \frac{1}{5 \times 6} + \ldots + \frac{1}{52 \times 53}\). [4]
AQA FP2 2013 January Q3
7 marks Standard +0.3
  1. Show that \(\frac{1}{5r-2} - \frac{1}{5r+3} = \frac{A}{(5r-2)(5r+3)}\), stating the value of the constant \(A\). [2 marks]
  2. Hence use the method of differences to show that $$\sum_{r=1}^{n} \frac{1}{(5r-2)(5r+3)} = \frac{n}{3(5n+3)}$$ [4 marks]
  3. Find the value of $$\sum_{r=1}^{\infty} \frac{1}{(5r-2)(5r+3)}$$ [1 mark]
AQA FP2 2013 January Q7
9 marks Standard +0.8
The polynomial \(\text{p}(n)\) is given by \(\text{p}(n) = (n-1)^3 + n^3 + (n+1)^3\).
    1. Show that \(\text{p}(k+1) - \text{p}(k)\), where \(k\) is a positive integer, is a multiple of 9. [3 marks]
    2. Prove by induction that \(\text{p}(n)\) is a multiple of 9 for all integers \(n \geqslant 1\). [4 marks]
  2. Using the result from part (a)(ii), show that \(n(n^2 + 2)\) is a multiple of 3 for any positive integer \(n\). [2 marks]
AQA FP2 2011 June Q3
6 marks Standard +0.8
  1. Show that $$(r + 1)! - (r - 1)! = (r^2 + r - 1)(r - 1)!$$ [2 marks]
  2. Hence show that $$\sum_{r=1}^{n} (r^2 + r - 1)(r - 1)! = (n + 2)n! - 2$$ [4 marks]
AQA FP2 2016 June Q1
6 marks Standard +0.3
  1. Given that \(f(r) = \frac{1}{4r-1}\), show that $$f(r) - f(r+1) = \frac{A}{(4r-1)(4r+3)}$$ where \(A\) is an integer. [2 marks]
  2. Use the method of differences to find the value of \(\sum_{r=1}^{50} \frac{1}{(4r-1)(4r+3)}\), giving your answer as a fraction in its simplest form. [4 marks]
OCR FP2 2012 January Q9
11 marks Challenging +1.3
  1. Show that \(\tanh(\ln n) = \frac{n^2 - 1}{n^2 + 1}\). [2]
It is given that, for non-negative integers \(n\), \(I_n = \int_0^{\ln 2} \tanh^n u du\).
  1. Show that \(I_n - I_{n-2} = -\frac{1}{n-1}\left(\frac{3}{5}\right)^{n-1}\), for \(n \geq 2\). [3]
  2. Find the value of \(I_3\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants. [4]
  3. Use the method of differences on the result of part (ii) to find the sum of the infinite series $$\frac{1}{2}\left(\frac{3}{5}\right)^2 + \frac{1}{4}\left(\frac{3}{5}\right)^4 + \frac{1}{6}\left(\frac{3}{5}\right)^6 + \ldots.$$ [2]
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]
AQA Further AS Paper 1 2018 June Q15
4 marks Standard +0.3
  1. Show that $$\frac{1}{r + 2} - \frac{1}{r + 3} = \frac{1}{(r + 2)(r + 3)}$$ [1 mark]
  2. Use the method of differences to show that $$\sum_{r=1}^{n} \frac{1}{(r + 2)(r + 3)} = \frac{n}{3(n + 3)}$$ [3 marks]
AQA Further AS Paper 1 2019 June Q7
5 marks Standard +0.8
  1. Show that $$\frac{1}{r-1} - \frac{1}{r+1} = \frac{A}{r^2-1}$$ where \(A\) is a constant to be found. [1 mark]
  2. Hence use the method of differences to show that $$\sum_{r=2}^n \frac{1}{r^2-1} = \frac{an^2 + bn + c}{4n(n+1)}$$ where \(a\), \(b\) and \(c\) are integers to be found. [4 marks]
AQA Further AS Paper 1 2020 June Q5
4 marks Standard +0.3
  1. Show that $$r^2(r + 1)^2 - (r - 1)^2r^2 = pr^3$$ where \(p\) is an integer to be found. [1 mark]
  2. Hence use the method of differences to show that $$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n + 1)^2$$ [3 marks]
AQA Further Paper 1 2023 June Q5
6 marks Standard +0.8
The function f is defined by $$f(r) = 2^r(r - 2) \quad (r \in \mathbb{Z})$$
  1. Show that $$f(r + 1) - f(r) = r2^r$$ [2 marks]
  2. Use the method of differences to show that $$\sum_{r=1}^n r2^r = 2^{n+1}(n - 1) + 2$$ [4 marks]
AQA Further Paper 1 Specimen Q3
6 marks Standard +0.8
\begin{enumerate}[label=(\alph*)] \item Given that $$\frac{2}{(r + 1)(r + 2)(r + 3)} \equiv \frac{A}{(r + 1)(r + 2)} + \frac{B}{(r + 2)(r + 3)}$$ find the values of the integers \(A\) and \(B\) [2 marks] \item Use the method of differences to show clearly that $$\sum_{r=4}^{97} \frac{1}{(r + 1)(r + 2)(r + 3)} = \frac{89}{19800}$$ [4 marks]
AQA Further Paper 2 2019 June Q14
12 marks Challenging +1.8
Let $$S_n = \sum_{r=1}^{n} \frac{1}{(r+1)(r+3)}$$ where \(n \geq 1\)
  1. Use the method of differences to show that $$S_n = \frac{5n^2 + an}{12(n+b)(n+c)}$$ where \(a\), \(b\) and \(c\) are integers. [6 marks]
  2. Show that, for any number \(k\) greater than \(\frac{12}{5}\), if the difference between \(\frac{5}{12}\) and \(S_n\) is less than \(\frac{1}{k}\), then $$n > \frac{k-5+\sqrt{k^2+1}}{2}$$ [6 marks]
AQA Further Paper 2 2024 June Q5
3 marks Standard +0.3
The first four terms of the series \(S\) can be written as $$S = (1 \times 2) + (2 \times 3) + (3 \times 4) + (4 \times 5) + ...$$
  1. Write an expression, using \(\sum\) notation, for the sum of the first \(n\) terms of \(S\) [1 mark]
  2. Show that the sum of the first \(n\) terms of \(S\) is equal to $$\frac{1}{3}n(n + 1)(n + 2)$$ [2 marks]
AQA Further Paper 2 2024 June Q13
8 marks Standard +0.8
  1. Use the method of differences to show that $$\sum_{r=2}^{n} \frac{1}{(r - 1)r(r + 1)} = \frac{1}{4} - \frac{1}{2n} + \frac{1}{2(n + 1)}$$ [5 marks]
  2. Find the smallest integer \(n\) such that $$\sum_{r=2}^{n} \frac{1}{(r - 1)r(r + 1)} > 0.24999$$ [3 marks]
OCR Further Pure Core 1 2021 November Q10
8 marks Challenging +1.2
Using an algebraic method, determine the least value of \(n\) for which \(\sum_{r=1}^{n} \frac{1}{(2r-1)(2r+1)} \geqslant 0.49\). [8]
OCR Further Pure Core 2 2024 June Q1
5 marks Moderate -0.8
  1. Use the method of differences to show that \(\sum_{r=1}^{n}\left(\frac{1}{r} - \frac{1}{r+1}\right) = 1 - \frac{1}{n+1}\). [1]
  2. Hence determine the following sums.
    1. \(\sum_{r=1}^{90}\frac{1}{r} - \frac{1}{r+1}\) [1]
    2. \(\sum_{r=100}^{\infty}\frac{1}{r} - \frac{1}{r+1}\) [3]
OCR Further Pure Core 2 Specimen Q3
4 marks Standard +0.3
\begin{enumerate}[label=(\roman*)] \item Find \(\sum_{r=1}^{n}\left(\frac{1}{r}-\frac{1}{r+2}\right)\). [3] \item What does the sum in part (i) tend to as \(n \to \infty\)? Justify your answer. [1]
OCR MEI Further Pure Core AS 2018 June Q7
9 marks Standard +0.8
  1. Express \(\frac{1}{2r-1} - \frac{1}{2r+1}\) as a single fraction. [2]
  2. Find how many terms of the series $$\frac{2}{1 \times 3} + \frac{2}{3 \times 5} + \frac{2}{5 \times 7} + \ldots + \frac{2}{(2r-1)(2r+1)} + \ldots$$ are needed for the sum to exceed \(0.999999\). [7]
OCR MEI Further Pure Core Specimen Q5
7 marks Standard +0.8
  1. Express \(\frac{2}{(r+1)(r+3)}\) in partial fractions. [2]
  2. Hence find \(\sum_{r=1}^{n} \frac{1}{(r+1)(r+3)}\), expressing your answer as a single fraction. [5]
WJEC Further Unit 1 2018 June Q5
8 marks Standard +0.3
  1. Show that \(\frac{2}{n-1} - \frac{2}{n+1}\) can be expressed as \(\frac{4}{(n^2-1)}\). [1]
  2. Hence, find an expression for \(\sum_{r=2}^{n} \frac{4}{(r^2-1)}\) in the form \(\frac{(an+b)(n+c)}{n(n+1)}\), where \(a\), \(b\), \(c\) are integers whose values are to be determined. [6]
  3. Explain why \(\sum_{r=1}^{100} \frac{4}{(r^2-1)}\) cannot be calculated. [1]
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 Q1
7 marks Standard +0.3
In this question you must show detailed reasoning.
  1. By using partial fractions show that \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2} = \frac{1}{2} - \frac{1}{n+2}\). [5]
  2. Hence determine the value of \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2}\). [2]