A-level · Sequences & series

Sigma notation and recurrence relationsExam-style questions

Reading and evaluating sums written with Σ, and generating sequences from a recurrence relation — including spotting periodic behaviour.

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27 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
The terms of a sequence $u_1, u_2, u_3, \ldots$ satisfy

$u_{n+1} = (-1)^{n}\left(u_n + k\right) \qquad n \geqslant 1$

where $k$ is a constant and $u_1 = 9$
(a) Show that $u_5 = 9$

(3 marks)

The sum of the first $42$ terms of the sequence is $63$
(b) Find the value of $k$ .

(2 marks)
(c) Hence find the value of

$\sum_{r=1}^{99} u_r$

(2 marks)
Level 4 · 7 marksTry it
2
The sequence $u_1, u_2, u_3, \ldots$ $u_{1}, u_{2}, u_{3}, \dots$ has first term $u_1 = 5$ $u_{1} = 5$ and satisfies

$u_{n+1} = au_n + b \qquad n \geqslant 1$

where $a$ $a$ and $b$ $b$ are constants.
(a) Find an expression, in terms of $a$ $a$ and $b$ $b$ , for

(i) $u_2$ $u_{2}$

(ii) $u_3$ $u_{3}$

(2 marks)

Given that
- $u_3 = 12$
- $3a + b = -6$
(b) show that

$2a^{2} - 9a - 18 = 0$

(3 marks)
(c) Hence find the smaller possible value of $u_4$ $u_{4}$

(3 marks)
Level 4 · 8 marksTry it
3
A sequence is defined by $u_1 = 3$ and $u_{n+1} = 2u_n - 1$ .
(a) Find $u_2$ , $u_3$ and $u_4$ . (3 marks)
(b) Evaluate $\displaystyle\sum_{n=1}^{4} u_n$ . (2 marks)
Level 4 · 5 marksTry it
4
A sequence is defined by $u_{1} = 3$ and $u_{n+1} = ku_{n} + 2$ , where $k$ is a constant.
(a) Find expressions for $u_{2}$ and $u_{3}$ in terms of $k$ . (2 marks)
(b) Given that $u_{3} = 10$ and that $k$ is an integer, find the value of $k$ . (3 marks)
Level 4 · 5 marksTry it
5
A sequence is defined by $u_{1} = 4$ and $u_{n+1} = \dfrac{2u_{n} - 3}{u_{n} - 1}$ .
(a) Show that the sequence is periodic, and state its period. (2 marks)
(b) Write down $u_{99}$ . (1 mark)
(c) Find $\displaystyle\sum_{n=1}^{90} u_{n}$ . (2 marks)
Level 4 · 5 marksTry it

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