1.04j Sum to infinity: convergent geometric series |r|<1

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CAIE P1 2024 November Q6
5 marks Standard +0.3
The first term of a convergent geometric progression is 10. The sum of the first 4 terms of the progression is \(p\) and the sum of the first 8 terms of the progression is \(q\). It is given that \(\frac{q}{p} = \frac{17}{16}\). Find the two possible values of the sum to infinity. [5]
CAIE P1 2010 June Q7
8 marks Moderate -0.8
  1. Find the sum of all the multiples of 5 between 100 and 300 inclusive. [3]
  2. A geometric progression has a common ratio of \(-\frac{2}{3}\) and the sum of the first 3 terms is 35. Find
    1. the first term of the progression, [3]
    2. the sum to infinity. [2]
CAIE P1 2011 June Q10
11 marks Standard +0.3
  1. A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is 5 cm, find the perimeter of the smallest sector. [6]
  2. The first, second and third terms of a geometric progression are \(2k + 3\), \(k + 6\) and \(k\), respectively. Given that all the terms of the geometric progression are positive, calculate
    1. the value of the constant \(k\), [3]
    2. the sum to infinity of the progression. [2]
CAIE P1 2015 June Q7
8 marks Moderate -0.3
  1. The third and fourth terms of a geometric progression are \(\frac{1}{4}\) and \(\frac{2}{9}\) respectively. Find the sum to infinity of the progression. [4]
  2. A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic progression. Given that the angle of the largest sector is 4 times the angle of the smallest sector, find the angle of the largest sector. [4]
CAIE P1 2015 June Q8
9 marks Moderate -0.8
  1. The first, second and last terms in an arithmetic progression are 56, 53 and \(-22\) respectively. Find the sum of all the terms in the progression. [4]
  2. The first, second and third terms of a geometric progression are \(2k + 6\), \(2k\) and \(k + 2\) respectively, where \(k\) is a positive constant.
    1. Find the value of \(k\). [3]
    2. Find the sum to infinity of the progression. [2]
CAIE P1 2017 June Q2
4 marks Standard +0.3
The common ratio of a geometric progression is \(r\). The first term of the progression is \((r^2 - 3r + 2)\) and the sum to infinity is \(S\).
  1. Show that \(S = 2 - r\). [2]
  2. Find the set of possible values that \(S\) can take. [2]
CAIE P1 2014 November Q8
8 marks Moderate -0.3
  1. The sum, \(S_n\), of the first \(n\) terms of an arithmetic progression is given by \(S_n = 32n - n^2\). Find the first term and the common difference. [3]
  2. A geometric progression in which all the terms are positive has sum to infinity 20. The sum of the first two terms is 12.8. Find the first term of the progression. [5]
CAIE P1 2014 November Q4
6 marks Standard +0.3
Three geometric progressions, \(P\), \(Q\) and \(R\), are such that their sums to infinity are the first three terms respectively of an arithmetic progression. Progression \(P\) is \(2, 1, \frac{1}{2}, \frac{1}{4}, \ldots\) Progression \(Q\) is \(3, 1, \frac{1}{3}, \frac{1}{9}, \ldots\)
  1. Find the sum to infinity of progression \(R\). [3]
  2. Given that the first term of \(R\) is 4, find the sum of the first three terms of \(R\). [3]
CAIE P1 2016 November Q9
8 marks Standard +0.3
  1. Two convergent geometric progressions, \(P\) and \(Q\), have the same sum to infinity. The first and second terms of \(P\) are \(6\) and \(6r\) respectively. The first and second terms of \(Q\) are \(12\) and \(-12r\) respectively. Find the value of the common sum to infinity. [3]
  2. The first term of an arithmetic progression is \(\cos\theta\) and the second term is \(\cos\theta + \sin^2\theta\), where \(0 \leq \theta \leq \pi\). The sum of the first \(13\) terms is \(52\). Find the possible values of \(\theta\). [5]
CAIE P3 2018 June Q3
5 marks Standard +0.3
The common ratio of a geometric progression is 0.99. Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures. [5]
Edexcel P2 2022 June Q6
8 marks Moderate -0.3
In a geometric sequence \(u_1, u_2, u_3, \ldots\)
  • the common ratio is \(r\)
  • \(u_2 + u_3 = 6\)
  • \(u_4 = 8\)
  1. Show that \(r\) satisfies $$3r^2 - 4r - 4 = 0$$ [3]
Given that the geometric sequence has a sum to infinity,
  1. find \(u_1\) [3]
  2. find \(S_∞\) [2]
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 Q4
11 marks Moderate -0.3
The first term of a geometric series is 120. The sum to infinity of the series is 480.
  1. Show that the common ratio, \(r\), is \(\frac{3}{4}\). [3]
  2. Find, to 2 decimal places, the difference between the 5th and 6th terms. [2]
  3. Calculate the sum of the first 7 terms. [2]
The sum of the first \(n\) terms of the series is greater than 300.
  1. Calculate the smallest possible value of \(n\). [4]
Edexcel C2 Q5
7 marks Moderate -0.8
The second and fifth terms of a geometric series are 9 and 1.125 respectively. For this series find
  1. the value of the common ratio, [3]
  2. the first term, [2]
  3. the sum to infinity. [2]
Edexcel C2 Q3
10 marks Moderate -0.8
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 Q16
12 marks Moderate -0.3
A geometric series is \(a + ar + ar^2 + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is given by $$S_n = \frac{a(1 - r^n)}{1 - r}.$$ [4]
The second and fourth terms of the series are 3 and 1.08 respectively. Given that all terms in the series are positive, find
  1. the value of \(r\) and the value of \(a\), [5]
  2. the sum to infinity of the series. [3]
Edexcel C2 Q37
10 marks Standard +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]
AQA C2 2009 June Q7
11 marks Moderate -0.3
A geometric series has second term \(375\) and fifth term \(81\).
    1. Show that the common ratio of the series is \(0.6\). [3]
    2. Find the first term of the series. [2]
  2. Find the sum to infinity of the series. [2]
  3. The \(n\)th term of the series is \(u_n\). Find the value of \(\sum_{n=6}^{\infty} u_n\). [4]
Edexcel C2 Q7
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]
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 Q6
12 marks Moderate -0.3
A geometric series is \(a + ar + ar^2 + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is given by $$S_n = \frac{a(1-r^n)}{1-r}$$ [4]
The second and fourth terms of the series are 3 and 1.08 respectively. Given that all terms in the series are positive, find
  1. the value of \(r\) and the value of \(a\), [5]
  2. the sum to infinity of the series. [3]
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]
Edexcel C2 Q8
12 marks Standard +0.3
A geometric series is \(a + ar + ar^2 + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is \(S_n = \frac{a(1 - r^n)}{1 - r}\). [4]
The first and second terms of a geometric series \(G\) are 10 and 9 respectively.
  1. Find, to 3 significant figures, the sum of the first twenty terms of \(G\). [3]
  2. Find the sum to infinity of \(G\). [2]
Another geometric series has its first term equal to its common ratio. The sum to infinity of this series is 10.
  1. Find the exact value of the common ratio of this series. [3]
OCR C2 Specimen Q4
7 marks Moderate -0.8
Records are kept of the number of copies of a certain book that are sold each week. In the first week after publication 3000 copies were sold, and in the second week 2400 copies were sold. The publisher forecasts future sales by assuming that the number of copies sold each week will form a geometric progression with first two terms 3000 and 2400. Calculate the publisher's forecasts for
  1. the number of copies that will be sold in the 20th week after publication, [3]
  2. the total number of copies sold during the first 20 weeks after publication, [2]
  3. the total number of copies that will ever be sold. [2]