Find sum to infinity - Geometric Sequences and Series

OCR MEI C2 2005 January Q5

5 marks Moderate -0.8

5 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2.
Find also the sum to infinity of this progression.

OCR MEI C2 2007 January Q2

3 marks Moderate -0.8

2 A geometric progression has 6 as its first term. Its sum to infinity is 5 .
Calculate its common ratio.

OCR MEI C2 2008 January Q8

5 marks Moderate -0.5

8 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.

OCR MEI C2 Q7

3 marks Moderate -0.8

7 The first two terms of a geometric series are 5 and 4.
Find

the sum of the first 10 terms,
the sum to infinity.

OCR MEI C2 Q3

3 marks Moderate -0.8

3 The sum to infinity of a geometric series is 5 and the first term is 2 .
Find the common ratio of the series.

OCR MEI C2 Q1

5 marks Moderate -0.5

1 The common ratio of a geometric progression is - 0.5 . The sum of its first three terms is 15 . Find the first term.
Find also the sum to infinity.

OCR C2 Q7

10 marks Standard +0.3

7. The second and third terms of a geometric series are $\log _ { 3 } 4$ and $\log _ { 3 } 16$ respectively.

Find the common ratio of the series.
Show that the first term of the series is $\log _ { 3 } 2$.
Find, to 3 significant figures, the sum of the first six terms of the series.

OCR C2 2009 January Q6

8 marks Moderate -0.8

6 A geometric progression has first term 20 and common ratio 0.9.

Find the sum to infinity.
Find the sum of the first 30 terms.
Use logarithms to find the smallest value of $p$ such that the $p$ th term is less than 0.4 .

OCR MEI Paper 2 2023 June Q1

3 marks Easy -1.2

1 Determine the sum of the infinite geometric series $9 - 3 + 1 - \frac { 1 } { 3 } + \frac { 1 } { 9 } + \ldots$

OCR MEI Paper 3 2020 November Q3

3 marks Moderate -0.8

3 A particular phone battery will last 10 hours when it is first used. Every time it is recharged, it will only last $98 \%$ of its previous time. Find the maximum total length of use for the battery.

AQA C2 2013 January Q6

10 marks Moderate -0.8

6

A geometric series begins $420 + 294 + 205.8 + \ldots$.
1. Show that the common ratio of the series is 0.7 .
2. Find the sum to infinity of the series.
3. Write the $n$th term of the series in the form $p \times q ^ { n }$, where $p$ and $q$ are constants.
The first term of an arithmetic series is 240 and the common difference of the series is - 8 . The $n$th term of the series is $u _ { n }$.
1. Write down an expression for $u _ { n }$.
2. Given that $u _ { k } = 0$, find the value of $\sum _ { n = 1 } ^ { k } u _ { n }$.

AQA C2 2008 June Q3

7 marks Moderate -0.8

3 A geometric series begins $$20 + 16 + 12.8 + 10.24 + \ldots$$

Find the common ratio of the series.
Find the sum to infinity of the series.
Find the sum of the first 20 terms of the series, giving your answer to three decimal places.
Prove that the $n$th term of the series is $25 \times 0.8 ^ { n }$.

AQA C2 2011 June Q9

10 marks Moderate -0.8

9 The first term of a geometric series is 12 and the common ratio of the series is $\frac { 3 } { 8 }$.

Find the sum to infinity of the series.
Show that the sixth term of the series can be written in the form $\frac { 3 ^ { 6 } } { 2 ^ { 13 } }$.
The $n$th term of the series is $u _ { n }$.
1. Write down an expression for $u _ { n }$ in terms of $n$.
2. Hence show that $$\log _ { a } u _ { n } = n \log _ { a } 3 - ( 3 n - 5 ) \log _ { a } 2$$ (4 marks)

AQA C2 2012 June Q4

8 marks Moderate -0.8

4 The $n$th term of a geometric series is $u _ { n }$, where $u _ { n } = 48 \left( \frac { 1 } { 4 } \right) ^ { n }$.

Find the value of $u _ { 1 }$ and the value of $u _ { 2 }$.
Find the value of the common ratio of the series.
Find the sum to infinity of the series.
Find the value of $\sum _ { n = 4 } ^ { \infty } u _ { n }$.

AQA C2 2013 June Q1

5 marks Easy -1.2

1 A geometric series has first term 80 and common ratio $\frac { 1 } { 2 }$.

Find the third term of the series.
Find the sum to infinity of the series.
Find the sum of the first 12 terms of the series, giving your answer to two decimal places.

AQA C2 2014 June Q3

6 marks Moderate -0.8

3 The first term of a geometric series is 54 and the common ratio of the series is $\frac { 8 } { 9 }$.

Find the sum to infinity of the series.
Find the second term of the series.
Show that the 12th term of the series can be written in the form $\frac { 2 ^ { p } } { 3 ^ { q } }$, where $p$ and $q$ are integers.
[0pt] [3 marks]

AQA C2 2015 June Q3

7 marks Moderate -0.8

3 The first term of an infinite geometric series is 48 . The common ratio of the series is 0.6 .

Find the third term of the series.
Find the sum to infinity of the series.
The $n$th term of the series is $u _ { n }$. Find the value of $\sum _ { n = 4 } ^ { \infty } u _ { n }$.

Pre-U Pre-U 9794/1 2012 June Q1

5 marks Easy -1.2

1 The first term of a geometric progression is 16 and the common ratio is 0.8 .

Calculate the sum of the first 12 terms.
Find the sum to infinity.

Pre-U Pre-U 9794/1 2012 Specimen Q3

6 marks Easy -1.2

3

In an arithmetic progression, the first term is 7 and the sum of the first 40 terms is 4960. Find the common difference.
A geometric progression has first term 14 and common ratio 0.3. Find the sum to infinity.

CAIE P1 2015 June Q7

8 marks Moderate -0.3

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]
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]

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

the value of the common ratio, [3]
the first term, [2]
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

the common ratio of the series, [2]
the first term of the series, [2]
the sum to infinity of the series. [2]
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 Q37

10 marks Standard +0.3

A geometric series has first term 1200. Its sum to infinity is 960.

Show that the common ratio of the series is $-\frac{1}{4}$. [3]
Find, to 3 decimal places, the difference between the ninth and tenth terms of the series. [3]
Write down an expression for the sum of the first $n$ terms of the series. [2]

Given that $n$ is odd,

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]
Find the sum to infinity of the series. [2]
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 $960$.

Show that the common ratio of the series is $-\frac{1}{4}$. [3]
Find, to 3 decimal places, the difference between the ninth and tenth terms of the series. [3]
Write down an expression for the sum of the first $n$ terms of the series. [2]

Given that $n$ is odd,

prove that the sum of the first $n$ terms of the series is $$960(1 + 0.25^n).$$ [2]