Find common ratio from terms

A question is this type if and only if it requires finding the common ratio of a geometric progression given two or more specific terms (e.g., second and fourth terms, or third and sixth terms).

9 questions · Moderate -0.5

1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1
Sort by: Default | Easiest first | Hardest first
CAIE P1 2009 November Q3
6 marks Easy -1.2
3 A progression has a second term of 96 and a fourth term of 54. Find the first term of the progression in each of the following cases:
  1. the progression is arithmetic,
  2. the progression is geometric with a positive common ratio.
AQA C2 2011 January Q6
9 marks Moderate -0.8
6 A geometric series has third term 36 and sixth term 972.
    1. Show that the common ratio of the series is 3 .
    2. Find the first term of the series.
  2. The \(n\)th term of the series is \(u _ { n }\).
    1. Show that \(\sum _ { n = 1 } ^ { 20 } u _ { n } = 2 \left( 3 ^ { 20 } - 1 \right)\).
    2. Find the least value of \(n\) such that \(u _ { n } > 4 \times 10 ^ { 15 }\). \(7 \quad\) A curve \(C\) is defined for \(x > 0\) by the equation \(y = x + 3 + \frac { 8 } { x ^ { 4 } }\) and is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-08_602_799_447_632}
Edexcel C2 Q5
8 marks Moderate -0.8
5. A geometric series has third term 36 and fourth term 27. Find
  1. the common ratio of the series,
  2. the fifth term of the series,
  3. the sum to infinity of the series.
Edexcel C2 Q8
9 marks Moderate -0.3
8. The second and third terms of a geometric series are \(\log _ { 3 } 4\) and \(\log _ { 3 } 16\) respectively.
  1. Find the common ratio of the series.
  2. Show that the first term of the series is \(\log _ { 3 } 2\).
  3. Find, to 3 significant figures, the sum of the first six terms of the series.
AQA C2 2007 January Q5
7 marks Moderate -0.8
5 The second term of a geometric series is 48 and the fourth term is 3 .
  1. Show that one possible value for the common ratio, \(r\), of the series is \(- \frac { 1 } { 4 }\) and state the other value.
  2. In the case when \(r = - \frac { 1 } { 4 }\), find:
    1. the first term;
    2. the sum to infinity of the series.
OCR C2 Q9
11 marks Standard +0.3
9. 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.
  2. Find the sum to infinity of the series.
  3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2 ^ { 6 - n }\).
Edexcel C2 Q6
8 marks Moderate -0.3
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]
  2. the first term, [2]
  3. the sum of the first 50 terms, giving your answer to 3 decimal places, [2]
  4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places. [2]
Edexcel C2 Q2
6 marks Moderate -0.3
The fourth term of a geometric series is 10 and the seventh term of the series is 80. For this series, find
  1. the common ratio, [2]
  2. the first term, [2]
  3. the sum of the first 20 terms, giving your answer to the nearest whole number. [2]
Edexcel C2 2008 January Q2
6 marks Moderate -0.3
The fourth term of a geometric series is 10 and the seventh term of the series is 80. For this series, find
  1. the common ratio, [2]
  2. the first term, [2]
  3. the sum of the first 20 terms, giving your answer to the nearest whole number. [2]