1.04g Sigma notation: for sums of series

136 questions

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Pre-U Pre-U 9794/1 2016 June Q4
3 marks Easy -1.3
4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), is defined by \(u _ { n } = 3 n + 5\).
  1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. Find the value of \(n\) such that \(u _ { n } = 254\).
  3. Evaluate \(\sum _ { n = 1 } ^ { 500 } u _ { n }\).
Edexcel C1 Q4
5 marks Easy -1.2
A sequence \(a_1, a_2, a_3, \ldots\) is defined by $$a_1 = 3,$$ $$a_{n+1} = 3a_n - 5, \quad n \geq 1.$$
  1. Find the value \(a_2\) and the value of \(a_3\). [2]
  2. Calculate the value of \(\sum_{r=1}^5 a_r\). [3]
Edexcel C1 Q13
6 marks Moderate -0.3
  1. Find the sum of all the integers between 1 and 1000 which are divisible by 7. [3]
  2. Hence, or otherwise, evaluate \(\sum_{r=1}^{142} (7r + 2)\). [3]
Edexcel C1 Specimen Q1
3 marks Easy -1.2
Calculate \(\sum_{r=1}^{20} 5 + 2r\) [3]
Edexcel FP1 Q27
6 marks Standard +0.3
Prove that \(\sum_{r=1}^{n} (r - 1)(r + 2) = \frac{1}{3} (n - 1)n(n + 4)\). [6]
Edexcel FP1 Q32
5 marks Standard +0.3
Prove by induction that, for \(n \in \mathbb{Z}^+\), \(\sum_{r=1}^{n} r 2^r = 2\{1 + (n - 1)2^n\}\). [5]
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]
Edexcel M2 2014 January Q5
5 marks Moderate -0.8
Given that for all positive integers \(n\), $$\sum_{r=1}^{n} a_r = 12 + 4n^2$$
  1. find the value of \(\sum_{r=1}^{5} a_r\) [2]
  2. Find the value of \(a_6\) [3]
Edexcel C1 Q1
6 marks Moderate -0.3
  1. Find the sum of all the integers between 1 and 1000 which are divisible by 7. [3]
  2. Hence, or otherwise, evaluate \(\sum_{r=1}^{142}(7r + 2)\). [3]
Edexcel C1 Q2
4 marks Moderate -0.5
Evaluate $$\sum_{r=10}^{30} (7 + 2r).$$ [4]
Edexcel C1 Q1
3 marks Easy -1.2
Evaluate $$\sum_{r=1}^{20} (3r + 4).$$ [3]
Edexcel C1 Q6
7 marks Moderate -0.8
  1. Evaluate $$\sum_{r=1}^{50} (80 - 3r).$$ [3]
  2. Show that $$\sum_{r=1}^{n} \frac{r + 3}{2} = k n(n + 7),$$ where \(k\) is a rational constant to be found. [4]
OCR C2 Q1
6 marks Easy -1.2
A sequence \(S\) has terms \(u_1, u_2, u_3, \ldots\) defined by $$u_n = 3n - 1,$$ for \(n \geqslant 1\).
  1. Write down the values of \(u_1, u_2\) and \(u_3\), and state what type of sequence \(S\) is. [3]
  2. Evaluate \(\sum_{n=1}^{100} u_n\). [3]
OCR MEI C2 2010 June Q2
4 marks Moderate -0.8
  1. Evaluate \(\sum_{r=2}^{5} \frac{1}{r-1}\). [2]
  2. Express the series \(2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7\) in the form \(\sum_{r=2}^{a} f(r)\) where \(f(r)\) and \(a\) are to be determined. [2]
OCR MEI C2 2014 June Q2
5 marks Moderate -0.8
  1. Find \(\sum_{r=1}^{5} \frac{21}{r+2}\). [2]
  2. A sequence is defined by $$u_1 = a, \text{ where } a \text{ is an unknown constant,}$$ $$u_{n+1} = u_n + 5.$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence. [3]
OCR MEI C2 2016 June Q2
3 marks Moderate -0.8
A sequence is defined as follows. \(u_1 = a\), where \(a > 0\) To obtain \(u_{r+1}\)
  • find the remainder when \(u_r\) is divided by 3,
  • multiply the remainder by 5,
  • the result is \(u_{r+1}\).
Find \(\sum_{r=2}^4 u_r\) in each of the following cases.
  1. \(a = 5\)
  2. \(a = 6\) [3]
Edexcel C2 Q7
9 marks Standard +0.3
  1. Prove that the sum of the first \(n\) terms of a geometric series with first term \(a\) and common ratio \(r\) is given by $$\frac{a(1-r^n)}{1-r}.$$ [4]
  2. Evaluate \(\sum_{r=1}^{12} (5 \times 2^r)\). [5]
OCR C2 Q7
10 marks Moderate -0.3
  1. Evaluate $$\sum_{r=10}^{30} (7 + 2r).$$ [4]
    1. Write down the formula for the sum of the first \(n\) positive integers. [1]
    2. Using this formula, find the sum of the integers from 100 to 200 inclusive. [3]
    3. Hence, find the sum of the integers between 300 and 600 inclusive which are divisible by 3. [2]
OCR MEI C2 Q3
5 marks Moderate -0.8
  1. Find \(\sum_{r=1}^{5} \frac{21}{r+2}\). [2]
  2. A sequence is defined by $$u_1 = a, \text{ where } a \text{ is an unknown constant,}$$ $$u_{n+1} = u_n + 5.$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence. [3]
AQA FP1 2016 June Q5
9 marks Standard +0.8
  1. Use the formulae for \(\sum_{r=1}^n r^2\) and \(\sum_{r=1}^n r\) to show that \(\sum_{r=1}^n (6r - 3)^2 = 3n(4n^2 - 1)\). [5 marks]
  2. Hence express \(\sum_{r=1}^{2n} r^3 - \sum_{r=1}^n (6r - 3)^2\) as a product of four linear factors in terms of \(n\). [4 marks]
Edexcel AEA 2004 June Q2
10 marks Challenging +1.3
  1. For the binomial expansion of \(\frac{1}{(1-x)^2}\), \(|x| < 1\), in ascending powers of \(x\),
    1. find the first four terms,
    2. write down the coefficient of \(x^n\). [2]
  2. Hence, show that, for \(|x| < 1\), \(\sum_{n=1}^{\infty} nx^n = \frac{x}{(1-x)^2}\). [2]
  3. Prove that, for \(|x| < 1\), \(\sum_{n=1}^{\infty} (an+1)x^n = \frac{(a+1)x-x^2}{(1-x)^2}\), where \(a\) is a constant. [4]
  4. Hence evaluate \(\sum_{n=1}^{\infty} \frac{5n+1}{2^{3n}}\). [2]
AQA Paper 1 2019 June Q8
4 marks Standard +0.3
$$P(n) = \sum_{k=0}^{n} k^3 - \sum_{k=0}^{n-1} k^3 \text{ where } n \text{ is a positive integer.}$$
  1. Find P(3) and P(10) [2 marks]
  2. Solve the equation \(P(n) = 1.25 \times 10^8\) [2 marks]
AQA Further Paper 2 2019 June Q4
3 marks Standard +0.3
The positive integer \(k\) is such that $$\sum_{r=1}^{k} (3r - k) = 90$$ Find the value of \(k\). [3 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]