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.

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

1
A sequence is defined by $u_{1} = 5$ and $u_{n+1} = u_{n} + 4$ .
(a) Write down $u_{2}$ , $u_{3}$ and $u_{4}$ . (2 marks)
(b) Find the value of $n$ for which $u_{n} = 101$ . (2 marks)
Level 2 · 4 marksTry it
2
A sequence $u_1, u_2, u_3, \ldots$ is defined by

$u_1 = 3$

$u_{n+1} = \frac{u_n + 1}{u_n - 1}$
(a) (i) Show that the sequence is periodic.

(ii) Write down the order of the sequence.

(2 marks)
(b) Find the value of $\displaystyle\sum_{n=1}^{75} u_n$

(2 marks)
Level 2 · 4 marksTry it
3
A sequence $u_1, u_2, u_3, \ldots$ is defined by

$u_1 = k$

$u_{n+1} = 8 - u_n$

where $k$ is a constant.
(a) Find, in terms of $k$ ,

(i) $u_2$

(ii) $u_3$

(2 marks)

Given that

$\sum_{n=1}^{41} u_n = 173$
(b) find the value of $k$ .

(2 marks)
Level 2 · 4 marksTry it
4
Evaluate $\displaystyle\sum_{r=1}^{10} (2r + 1)$ .
Level 3 · 2 marksTry it
5
A sequence is defined by $u_{n+1} = 2u_n - 1$ with $u_1 = 3$ . Find $u_2$ , $u_3$ and $u_4$ .
Level 3 · 2 marksTry it

22 more questions on sigma notation and recurrence relations

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