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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CAIE P1 2022 June Q2
4 marks Moderate -0.5
2 The second and third terms of a geometric progression are 10 and 8 respectively.
Find the sum to infinity.
CAIE P1 2023 June Q6
5 marks Standard +0.3
6 The first three terms of an arithmetic progression are \(\frac { p ^ { 2 } } { 6 } , 2 p - 6\) and \(p\).
  1. Given that the common difference of the progression is not zero, find the value of \(p\).
  2. Using this value, find the sum to infinity of the geometric progression with first two terms \(\frac { p ^ { 2 } } { 6 }\) and \(2 p - 6\).
CAIE P1 2023 November Q7
7 marks Standard +0.3
7 The sum of the first two terms of a geometric progression is 15 and the sum to infinity is \(\frac { 125 } { 7 }\). The common ratio of the progression is negative. Find the third term of the progression.
CAIE P1 2004 June Q1
4 marks Easy -1.2
1 A geometric progression has first term 64 and sum to infinity 256. Find
  1. the common ratio,
  2. the sum of the first ten terms.
CAIE P1 2009 June Q7
7 marks Moderate -0.8
7
  1. Find the sum to infinity of the geometric progression with first three terms \(0.5,0.5 ^ { 3 }\) and \(0.5 ^ { 5 }\).
  2. The first two terms in an arithmetic progression are 5 and 9. The last term in the progression is the only term which is greater than 200 . Find the sum of all the terms in the progression.
CAIE P1 2010 June Q1
5 marks Moderate -0.8
1 The first term of a geometric progression is 12 and the second term is - 6 . Find
  1. the tenth term of the progression,
  2. the sum to infinity.
CAIE P1 2013 June Q4
6 marks Moderate -0.5
4 The third term of a geometric progression is - 108 and the sixth term is 32 . Find
  1. the common ratio,
  2. the first term,
  3. the sum to infinity.
CAIE P1 2016 June Q9
9 marks Moderate -0.3
9
  1. The first term of a geometric progression in which all the terms are positive is 50 . The third term is 32 . Find the sum to infinity of the progression.
  2. The first three terms of an arithmetic progression are \(2 \sin x , 3 \cos x\) and ( \(\sin x + 2 \cos x\) ) respectively, where \(x\) is an acute angle.
    1. Show that \(\tan x = \frac { 4 } { 3 }\).
    2. Find the sum of the first twenty terms of the progression.
CAIE P1 2002 November Q2
5 marks Moderate -0.8
2 A geometric progression, for which the common ratio is positive, has a second term of 18 and a fourth term of 8 . Find
  1. the first term and the common ratio of the progression,
  2. the sum to infinity of the progression.
CAIE P1 2006 November Q6
9 marks Moderate -0.8
6
  1. Find the sum of all the integers between 100 and 400 that are divisible by 7 .
  2. The first three terms in a geometric progression are \(144 , x\) and 64 respectively, where \(x\) is positive. Find
    1. the value of \(x\),
    2. the sum to infinity of the progression.
CAIE P1 2010 November Q6
7 marks Moderate -0.3
6
  1. The fifth term of an arithmetic progression is 18 and the sum of the first 5 terms is 75 . Find the first term and the common difference.
  2. The first term of a geometric progression is 16 and the fourth term is \(\frac { 27 } { 4 }\). Find the sum to infinity of the progression.
CAIE P1 2012 November Q8
9 marks Standard +0.3
8
  1. In a geometric progression, all the terms are positive, the second term is 24 and the fourth term is \(13 \frac { 1 } { 2 }\). Find
    1. the first term,
    2. the sum to infinity of the progression.
  2. A circle is divided into \(n\) sectors in such a way that the angles of the sectors are in arithmetic progression. The smallest two angles are \(3 ^ { \circ }\) and \(5 ^ { \circ }\). Find the value of \(n\).
CAIE P1 2012 November Q5
5 marks Moderate -0.8
5 The first term of a geometric progression is \(5 \frac { 1 } { 3 }\) and the fourth term is \(2 \frac { 1 } { 4 }\). Find
  1. the common ratio,
  2. the sum to infinity.
CAIE P1 2016 November Q5
6 marks Standard +0.3
5 The sum of the 1st and 2nd terms of a geometric progression is 50 and the sum of the 2nd and 3rd terms is 30 . Find the sum to infinity.
CAIE P1 2018 November Q4
5 marks Easy -1.2
4 The first term of a series is 6 and the second term is 2 .
  1. For the case where the series is an arithmetic progression, find the sum of the first 80 terms.
  2. For the case where the series is a geometric progression, find the sum to infinity.
Edexcel P2 2022 October Q8
7 marks Moderate -0.3
  1. A geometric sequence has first term \(a\) and common ratio \(r\)
Given that \(S _ { \infty } = 3 a\)
  1. show that \(r = \frac { 2 } { 3 }\) Given also that $$u _ { 2 } - u _ { 4 } = 16$$ where \(u _ { k }\) is the \(k ^ { \text {th } }\) term of this sequence,
  2. find the value of \(S _ { 10 }\) giving your answer to one decimal place.
Edexcel C2 2005 January Q6
8 marks Moderate -0.3
  1. The second and fourth terms of a geometric series are 7.2 and 5.832 respectively.
The common ratio of the series is positive.
For this series, find
  1. the common ratio,
  2. the first term,
  3. the sum of the first 50 terms, giving your answer to 3 decimal places,
  4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places.
Edexcel C2 2011 January Q3
7 marks Moderate -0.3
3. The second and fifth terms of a geometric series are 750 and - 6 respectively. Find
  1. the common ratio of the series,
  2. the first term of the series,
  3. the sum to infinity of the series.
Edexcel C2 2012 January Q1
6 marks Easy -1.2
  1. A geometric series has first term \(a = 360\) and common ratio \(r = \frac { 7 } { 8 }\)
Giving your answers to 3 significant figures where appropriate, find
  1. the 20 th term of the series,
  2. the sum of the first 20 terms of the series,
  3. the sum to infinity of the series.
Edexcel C2 2009 June Q5
9 marks Moderate -0.3
  1. The third term of a geometric sequence is 324 and the sixth term is 96
    1. Show that the common ratio of the sequence is \(\frac { 2 } { 3 }\)
    2. Find the first term of the sequence.
    3. Find the sum of the first 15 terms of the sequence.
    4. Find the sum to infinity of the sequence.
Edexcel C2 2011 June Q6
10 marks Moderate -0.8
  1. The second and third terms of a geometric series are 192 and 144 respectively.
For this series, find
  1. the common ratio,
  2. the first term,
  3. the sum to infinity,
  4. the smallest value of \(n\) for which the sum of the first \(n\) terms of the series exceeds 1000.
Edexcel C2 2013 June Q1
4 marks Moderate -0.8
  1. The first three terms of a geometric series are
$$18,12 \text { and } p$$ respectively, where \(p\) is a constant. Find
  1. the value of the common ratio of the series,
  2. the value of \(p\),
  3. the sum of the first 15 terms of the series, giving your answer to 3 decimal places.
Edexcel C2 2015 June Q5
10 marks Standard +0.3
    1. All the terms of a geometric series are positive. The sum of the first two terms is 34 and the sum to infinity is 162
Find
  1. the common ratio,
  2. the first term.
    (ii) A different geometric series has a first term of 42 and a common ratio of \(\frac { 6 } { 7 }\). Find the smallest value of \(n\) for which the sum of the first \(n\) terms of the series exceeds 290
Edexcel C2 2016 June Q1
7 marks Moderate -0.8
  1. A geometric series has first term \(a\) and common ratio \(r = \frac { 3 } { 4 }\)
The sum of the first 4 terms of this series is 175
  1. Show that \(a = 64\)
  2. Find the sum to infinity of the series.
  3. Find the difference between the 9th and 10th terms of the series. Give your answer to 3 decimal places.
Edexcel C2 2018 June Q6
7 marks Moderate -0.8
  1. A geometric series with common ratio \(r = - 0.9\) has sum to infinity 10000 For this series,
    1. find the first term,
    2. find the fifth term,
    3. find the sum of the first twelve terms, giving this answer to the nearest integer.