Find sum to infinity

A question is this type if and only if it asks to find the sum to infinity of a geometric progression given sufficient information (e.g., specific terms, first term and common ratio, or relationships between terms).

63 questions · Moderate -0.6

1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1
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Edexcel C2 Q6
10 marks Moderate -0.3
The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find
  1. the common ratio of the series, [2]
  2. the first term of the series, [2]
  3. the sum to infinity of the series. [2]
  4. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series. [4]
Edexcel C2 Q5
10 marks Moderate -0.3
The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find
  1. the common ratio of the series, [2]
  2. the first term of the series, [2]
  3. the sum to infinity of the series. [2]
  4. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series. [4]
Edexcel C2 Q6
10 marks Moderate -0.3
A geometric series has first term \(1200\). Its sum to infinity is \(960\).
  1. Show that the common ratio of the series is \(-\frac{1}{4}\). [3]
  2. Find, to 3 decimal places, the difference between the ninth and tenth terms of the series. [3]
  3. Write down an expression for the sum of the first \(n\) terms of the series. [2]
Given that \(n\) is odd,
  1. prove that the sum of the first \(n\) terms of the series is \(960(1 + 0.25^n)\). [2]
OCR MEI C2 2008 June Q2
3 marks Easy -1.3
The first term of a geometric series is 5.4 and the common ratio is 0.1.
  1. Find the fourth term of the series. [1]
  2. Find the sum to infinity of the series. [2]
Edexcel C2 Q9
12 marks Standard +0.3
The second and fifth terms of a geometric series are \(-48\) and \(6\) respectively.
  1. Find the first term and the common ratio of the series. [5]
  2. Find the sum to infinity of the series. [2]
  3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2^{6-n}\). [5]
Edexcel C2 Q1
4 marks Easy -1.2
A geometric series has first term 75 and second term \(-15\).
  1. Find the common ratio of the series. [2]
  2. Find the sum to infinity of the series. [2]
OCR C2 Q4
8 marks Moderate -0.3
A geometric progression has third term 36 and fourth term 27. Find
  1. the common ratio, [2]
  2. the fifth term, [2]
  3. the sum to infinity. [4]
OCR C2 Q1
4 marks Easy -1.2
A geometric progression has first term 75 and second term \(-15\).
  1. Find the common ratio. [2]
  2. Find the sum to infinity. [2]
OCR MEI C2 Q6
5 marks Easy -1.2
  1. Find the 51st term of the sequence given by $$u_1 = 5,$$ $$u_{n+1} = u_n + 4.$$ [3]
  2. Find the sum to infinity of the geometric progression which begins $$5 \quad 2 \quad 0.8 \quad \ldots$$ [2]
OCR MEI C2 Q8
5 marks Moderate -0.3
The second term of a geometric progression is 18 and the fourth term is 2. The common ratio is positive. Find the sum to infinity of this progression. [5]
OCR MEI Paper 2 2022 June Q2
2 marks Easy -1.2
Find the sum of the infinite series \(50 + 25 + 12.5 + 6.25 + \ldots\). [2]
SPS SPS SM Pure 2021 May Q8
12 marks Challenging +1.2
In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g_n\) and the \(n\)th term of an arithmetic progression is denoted by \(a_n\). It is given that \(g_1 = a_1 = 1 + \sqrt{5}\), \(g_2 = a_2\) and \(g_3 + a_3 = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2\sqrt{5}\). [12]
SPS SPS SM Pure 2022 June Q9
5 marks Standard +0.3
A geometric series has second term 16 and fourth term 8 All the terms of the series are positive. The \(n\)th term of the series is \(u_n\) Find the exact value of \(\sum_{n=5}^{\infty} u_n\) [5 marks]