Recurrence Relations — Higher Maths Practice Questions

1

A sequence is generated by the recurrence relation

\displaystyle u_{n+1}=ku_n+4,

where

\displaystyle k

is a constant.
Given

\displaystyle u_0=-1

and

\displaystyle u_1=7,

find the value of

\displaystyle k

and the value of

\displaystyle u_{3}.

2

A sequence is defined by the recurrence relation

\displaystyle u_{n+1}=mu_n+c,

where

\displaystyle m

and

\displaystyle c

are constants. The first three terms of the sequence are

\displaystyle 12,

\displaystyle 20

and

\displaystyle 26.

Find the values of

\displaystyle m

and

\displaystyle c.

Hence find the value of the fourth term in the sequence.

3

A sequence is generated by the recurrence relation

\displaystyle u_{n+1}=mu_n-3,

where

\displaystyle m

is a positive integer.
Given

\displaystyle u_2=11

and

\displaystyle u_4=35,

find the value of

\displaystyle m

and the value of

\displaystyle u_{3}.

4

A sequence is defined by the recurrence relation

\displaystyle u_{n+1}=\frac{2}{3} u_n+4,

with

\displaystyle u_1=6.

Explain why this sequence approaches a limit as

\displaystyle n \to \infty

and calculate this limit.

5

The population of Common Blue butterflies in a woodland area is observed to be declining by

\displaystyle 3.5\%

per year. To increase the population, scientists plan to release

\displaystyle 500

of this species within the woodland at the end of June each year. Let

\displaystyle u_n

represent the population at the beginning of July,

\displaystyle n

years after the first annual reintroduction. It is known that

\displaystyle u_n

and

\displaystyle u_{n+1}

satisfy the recurrence relation

\displaystyle u_{n+1}=au_{n}+b.

(a) State the values of

\displaystyle a

and

\displaystyle b.

(b) Explain whether or not the population will stabilise in the long term.
(c) The population at the beginning of the reintroduction programme was estimated at

\displaystyle 10,000.

Explain whether or not the population will ever exceed

\displaystyle 15,000.

6

Sequences may be generated by recurrence relations of the form

\displaystyle u_{n+1}=ku_n-5,

\displaystyle u_0=20,

where

\displaystyle k \in \mathbb{R}.

(a) Show that

\displaystyle u_2=20k^2-5k-5.

(b) Find the range of values of

\displaystyle k

for which

\displaystyle u_2 < u_0.

7

Sequences may be defined by the linear recurrence relation

\displaystyle u_{n+1}=(3-k)u_n-2.

Find the range of values of

\displaystyle k

for which such a sequence converges to a limit.

8

2016 P1 Q3

(1, 1, 2)4 Marks

A sequence is defined by the recurrence relation
$\displaystyle u_{n+1}=\frac{1}{3}u_{n}+10$
with $\displaystyle u_{3}=6.$

(a)  Find the value of $\displaystyle u_{4}.$
(b)  Explain why this sequence approaches a limit as $\displaystyle n\rightarrow\infty.$
(c)  Calculate this limit.

9

2017 P1 Q9

(2, 1, 2)5 Marks

A sequence is generated by the recurrence relation
$\displaystyle u_{n+1}=mu_{n}+6$
where $\displaystyle m$ is a constant.

(a)  Given $\displaystyle u_{1}=28$ and $\displaystyle u_{2}=13$ , find the value of $\displaystyle m$ .
(b)  (i) Explain why this sequence approaches a limit as $\displaystyle n\rightarrow\infty$ .
      (ii) Calculate this limit.

10

2019 P1 Q4

(3, 1)4 Marks

A sequence is generated by the recurrence relation $\displaystyle u_{n+1}=mu_{n}+c$ , where the first three terms of the sequence are 6, 9 and 11.

(a) Find the values of $\displaystyle m$ and $\displaystyle c.$
(b) Hence, calculate the fourth term of the sequence.

11

2021 P1 Q13

A sequence is generated by the recurrence relation

\displaystyle u_{n+1}=\frac{2}{3}u_n+8,

\displaystyle u_7=20.

(a) Determine the value of

\displaystyle u_5.

(b) Determine the limit of this sequence.

12

2021 P2 Q12

A sequence of real numbers is such that the terms of the sequence satisfy the recurrence relation

\displaystyle u_{n+1}=9u_n-440

and

\displaystyle u_{n+1} > u_n

for all values of

\displaystyle n.

The difference between two particular terms,

\displaystyle u_{k+1}

and

\displaystyle u_k,

is

\displaystyle 1000.

Determine, algebraically, the value of

\displaystyle u_k.

13

2024 P1 Q2

(1, 1, 2)4 Marks

A sequence is defined by the recurrence relation
$\displaystyle u_{n+1}=\frac{1}{5}u_{n}+12$
with $\displaystyle u_{1}=20$
(a)  Calculate the value of $\displaystyle u_{2}.$
(b)  (i) Explain why this sequence approaches a limit as $\displaystyle n\rightarrow\infty$ .
      (ii) Calculate this limit.

14

2025 P2 Q9

(2, 1)3 Marks

A sequence satisfies the recurrence relation $\displaystyle u_{n+1}=mu_{n}+4$ , where m is a constant.
(a) The sequence approaches a limit of 10 as $\displaystyle n\rightarrow\infty.$ Determine the value of m.
(b) Given that $\displaystyle u_{1}=19$ , calculate the value of $\displaystyle u_{0}.$

15

2026 P2 Q5

(1, 1, 2)4 Marks

A sequence is defined by the recurrence relation $\displaystyle u_{n+1}=1.125u_{n}+5$ , $\displaystyle u_{0}=2.$
(a)  Calculate the value of $\displaystyle u_{1}.$
(b)  Explain why this sequence does not approach a limit as $\displaystyle n\rightarrow\infty.$
(c)  Determine the smallest value of n for which $\displaystyle u_{n} \gt 30.$