A-level · Sequences & series

Geometric sequences and seriesExam-style questions

Sequences with a constant common ratio — the nth term, the sum of n terms, and the sum to infinity (and exactly when it exists).

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29 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

An energy company begins fitting smart meters to homes in a city.

The numbers of smart meters fitted, in hundreds, in each of the first three months were

$k + 12 \qquad\qquad 2k + 4 \qquad\qquad 2k + 10$

respectively, where $k$ is a constant.

The company models the monthly numbers of smart meters fitted as consecutive terms of a geometric sequence.
(a) Show that $k^{2} - 9k - 52 = 0$

(2 marks)
(b) Find the number of smart meters that the model predicts will be fitted in the 4th month.

(4 marks)

According to the model, the total number of smart meters fitted first exceeds $60\,000$ in month $N$ .
(c) Find the value of $N$ .

(3 marks)
Level 4 · 9 marksTry it
2
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
(a) The second term of a geometric series is $72$ and the third term is $54$

(i) Find the sum to infinity of the series.

The sum of the first $n$ terms of the series is $S_n$

(ii) Using logarithms, find the smallest value of $n$ for which the sum to infinity of the series and $S_n$ differ by less than $0.1$

(5 marks)
(b) An arithmetic series has $n$ th term

$u_n = 157 - 6n$

Find the greatest value of $\displaystyle\sum_{r=1}^{n} u_r$

(4 marks)
Level 4 · 9 marksTry it
3
A ball is dropped from a height of $2$ m onto horizontal ground. After each bounce the ball rises to $65\%$ of the height it reached before that bounce.
(a) Find the height the ball reaches after the $5$ th bounce, giving your answer to $3$ significant figures. (2 marks)
(b) Find the total distance travelled by the ball from the moment it is dropped until the moment it hits the ground for the $6$ th time, giving your answer to $3$ significant figures. (4 marks)
(c) Assuming the ball continues to bounce indefinitely, find the total distance it travels, giving your answer to $3$ significant figures. (3 marks)
Level 4 · 9 marksTry it
4
A geometric series has first term $12$ and sum to infinity $16$ .
(a) Show that the common ratio of the series is $\dfrac{1}{4}$ . (3 marks)
(b) Find the smallest value of $n$ for which the sum to infinity differs from the sum of the first $n$ terms by less than $0.001$ . (5 marks)
Level 4 · 8 marksTry it
5
The first three terms of a geometric progression are $4k$ , $k+3$ and $k$ , where $k$ is a non-zero constant.
(a) Show that $k^{2} - 2k - 3 = 0$ and hence find the two possible values of $k$ . (4 marks)
(b) It is given that the common ratio of the progression is positive. Find the sum to infinity. (2 marks)
(c) An arithmetic progression has first term equal to the first term of the geometric progression in part (b), and common difference $5$ . Find the least number of terms whose sum exceeds $500$ . (3 marks)
Level 4 · 9 marksTry it

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