A-level · Sequences & series

Arithmetic sequences and seriesExam-style questions

Sequences with a constant common difference — the nth-term rule and the sum formula, and where each comes from.

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34 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
A regional airport records the number of flights that it handles each year.

The airport handled $6200$ flights in year 1 and expects to handle $17\,000$ flights in year 8.

In model A, the numbers of flights handled in years 1, 2, 3, … are taken to form an arithmetic sequence.
(a) Find, according to model A, the number of flights handled in year 5. Give your answer to the nearest 10.

(3 marks)

In model B, the numbers of flights handled in years 1, 2, 3, … are taken instead to form a geometric sequence.
(b) Find, according to model B, the number of flights handled in year 3. Give your answer to the nearest 10.

(3 marks)

The two models give different predictions for the total number of flights handled over years 1 to 8.
(c) Find the difference between these two totals. Give your answer to the nearest 10.

(3 marks)
Level 4 · 9 marksTry it
2
The sum of the first $n$ terms of a series is $S_{n} = 3n^{2} + 2n$ .
(a) Find the first term. (1 mark)
(b) Show that the series is arithmetic, and find an expression for the $n$ th term and the common difference. (4 marks)
Level 4 · 5 marksTry it
3
An arithmetic sequence has first term $a$ and non-zero common difference $d$ . The $2$ nd, $5$ th and $11$ th terms of the arithmetic sequence form the first three terms of a geometric sequence.
(a) Show that the common ratio of the geometric sequence is $2$ . (5 marks)
(b) Given that $a = 4$ , find the sum of the first $20$ terms of the arithmetic sequence. (2 marks)
Level 4 · 7 marksTry it
4
The first, second and fifth terms of an arithmetic sequence, with first term $a$ and common difference $d$ , where $d \neq 0$ , are also the first three terms of a geometric sequence.

The sum of the first $20$ terms of the arithmetic sequence is $800$ .

Find the smallest value of $n$ for which the $n$ th term of the geometric sequence is greater than one million. (6 marks)
Level 4 · 6 marksTry it
5
A charity receives a donation at the end of each week. The donation at the end of week $1$ is £ $120$ , and each week the donation is £ $45$ more than the donation the week before.
(a) Find the donation received at the end of week $30$ . (2 marks)
(b) The charity's target is a total of £ $50\,000$ . Find the number of the week at the end of which the running total first exceeds £ $50\,000$ . (3 marks)
Level 4 · 5 marksTry it

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