Sum from n+1 to 2n or similar range

A question is this type if and only if it requires finding a sum over a shifted or restricted range like Σ(r=n+1 to 2n) or Σ(r=n to n²) by subtracting two sums.

40 questions · Standard +0.7

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AQA FP1 2012 January Q4
7 marks Standard +0.3
4
  1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 4 r - 3 ) = k n ( n + 1 ) \left( 2 n ^ { 2 } - 1 \right)$$ where \(k\) is a constant.
  2. Hence evaluate $$\sum _ { r = 20 } ^ { 40 } r ^ { 2 } ( 4 r - 3 )$$ (2 marks)
AQA FP1 2013 January Q8
8 marks Standard +0.8
8
  1. Show that $$\sum _ { r = 1 } ^ { n } 2 r \left( 2 r ^ { 2 } - 3 r - 1 \right) = n ( n + p ) ( n + q ) ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
  2. Hence find the value of $$\sum _ { r = 11 } ^ { 20 } 2 r \left( 2 r ^ { 2 } - 3 r - 1 \right)$$ (2 marks)
AQA FP2 2010 January Q5
8 marks Standard +0.8
5 The sum to \(r\) terms, \(S _ { r }\), of a series is given by $$S _ { r } = r ^ { 2 } ( r + 1 ) ( r + 2 )$$ Given that \(u _ { r }\) is the \(r\) th term of the series whose sum is \(S _ { r }\), show that:
    1. \(u _ { 1 } = 6\);
    2. \(u _ { 2 } = 42\);
    3. \(\quad u _ { n } = n ( n + 1 ) ( 4 n - 1 )\).
  2. Show that $$\sum _ { r = n + 1 } ^ { 2 n } u _ { r } = 3 n ^ { 2 } ( n + 1 ) ( 5 n + 2 )$$
AQA FP1 2005 June Q3
7 marks Standard +0.3
3
  1. Use the formulae $$\begin{gathered} \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 ) \\ \sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 } \end{gathered}$$ and $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { 1 } { 12 } n \left( n ^ { 2 } - 1 \right) ( 3 n + 2 )$$ (4 marks)
  2. Use the result from part (a) to find the value of $$\sum _ { r = 4 } ^ { 11 } r ^ { 2 } ( r - 1 )$$ (3 marks)
AQA FP2 2009 January Q3
8 marks Challenging +1.2
3
  1. Given that \(\mathrm { f } ( r ) = \frac { 1 } { 4 } r ^ { 2 } ( r + 1 ) ^ { 2 }\), show that $$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = r ^ { 3 }$$
  2. Use the method of differences to show that $$\sum _ { r = n } ^ { 2 n } r ^ { 3 } = \frac { 3 } { 4 } n ^ { 2 } ( n + 1 ) ( 5 n + 1 )$$
AQA FP2 2007 June Q1
7 marks Standard +0.3
1
  1. Given that \(\mathrm { f } ( r ) = ( r - 1 ) r ^ { 2 }\), show that $$\mathrm { f } ( r + 1 ) - \mathrm { f } ( r ) = r ( 3 r + 1 )$$
  2. Use the method of differences to find the value of $$\sum _ { r = 50 } ^ { 99 } r ( 3 r + 1 )$$ (4 marks)
AQA Further AS Paper 1 2021 June Q9
7 marks Challenging +1.2
9
  1. Use the standard formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ( r + 3 ) = a n ( n + 1 ) ( n + b )$$ where \(a\) and \(b\) are constants to be determined.
    [0pt] [4 marks]
    9
  2. Hence, or otherwise, find a fully factorised expression for $$\sum _ { r = n + 1 } ^ { 5 n } r ( r + 3 )$$ $$\mathbf { A } = \left[ \begin{array} { c c } 3 & i - 1 \\ i & 2 \end{array} \right]$$
Pre-U Pre-U 9795/1 2014 June Q1
4 marks Standard +0.8
1 The series \(S\) is given by \(S = \sum _ { r = 0 } ^ { N } ( N + r ) ^ { 2 }\).
  1. Write out the first three terms and the last three terms of the series for \(S\).
  2. Use the standard result \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to show that \(S = \frac { 1 } { 6 } N ( N + 1 ) ( a N + 1 )\) for some positive integer \(a\) to be determined.
CAIE FP1 2003 November Q2
6 marks Challenging +1.2
Given that $$u_n = \frac{1}{n^2 - n + 1} - \frac{1}{n^2 + n + 1},$$ find \(S_N = \sum_{n=N+1}^{2N} u_n\) in terms of \(N\). [3] Find a number \(M\) such that \(S_N < 10^{-20}\) for all \(N > M\). [3]
Edexcel F1 2022 January Q9
14 marks Standard +0.8
  1. Prove by induction that, for \(n \in \mathbb{N}\) $$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2$$ [5]
  2. Using the standard summation formulae, show that $$\sum_{r=1}^{n} r(r+1)(r-1) = \frac{1}{4}n(n+A)(n+B)(n+C)$$ where \(A\), \(B\) and \(C\) are constants to be determined. [4]
  3. Determine the value of \(n\) for which $$3\sum_{r=1}^{n} r(r+1)(r-1) = 17\sum_{r=n}^{2n} r^2$$ [5]
Edexcel FP1 2013 June Q8
10 marks Standard +0.3
  1. Prove by induction, that for \(n \in \mathbb{Z}^+\), $$\sum_{r=1}^{n} r(2r - 1) = \frac{1}{6}n(n + 1)(4n - 1)$$ [6]
  2. Hence, show that $$\sum_{r=n+1}^{2n} r(2r - 1) = \frac{1}{3}n(an^2 + bn + c)$$ where \(a\), \(b\) and \(c\) are integers to be found. [4]
Edexcel FP2 Q6
21 marks Standard +0.3
  1. Express \(\frac{1}{r(r + 2)}\) in partial fractions. [2]
  2. Hence prove, by the method of differences, that $$\sum_{r=1}^{2n} \frac{1}{r(r + 2)} = \frac{n(4n + 5)}{4(n + 1)(n + 2)},$$ [6]
The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6| = 2|z - 3|.$$
  1. Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6]
where \(a\) and \(b\) are constants to be found.
  1. Hence show that $$\sum_{r=1}^{2n} \frac{1}{r(r + 2)} = \frac{n(4n + 5)}{4(n + 1)(n + 2)},$$ [3]
  2. Find the complex number for which both \(|z - 6| = 2|z - 3|\) and \(\arg(z - 6) = -\frac{3\pi}{4}\). [4]
Edexcel FP2 Q38
10 marks Standard +0.3
  1. Express \(\frac{1}{r(r + 2)}\) in partial fractions. [2]
  2. Hence prove, by the method of differences, that $$\sum_{r=1}^{n} \frac{4}{r(r + 2)} = \frac{n(3n + 5)}{(n + 1)(n + 2)}.$$ [5]
  3. Find the value of \(\sum_{r=50}^{100} \frac{4}{r(r + 2)}\), to 4 decimal places. [3]
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]
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]