Higher Maths · Worksheet 11

Recurrence Relations Worksheet | Higher Maths

Skills covered: Percentage change; linear recurrence formulae; computing terms; fixed limiting values; finding coefficients; financial and population models.

18 questions72 marks3 graded sectionsFree PDF

Download PDF Recurrence Relations revision notes

Instructions: Attempt all questions. Show sufficient working to justify your answers. Give exact values unless a degree of accuracy is stated. Each section is separate, so you can target the part you need.

These are the same worksheets our pupils use at the Pollokshields tuition centre. Start with Section A if recurrence relations still feels shaky; if you are confident, go straight to Section C and time yourself.

Section A

Prerequisite core skills

Revise earlier learning

Question 1

[3 marks]

An investment of

\pounds 2500

increases by

4\%

each year. Calculate its value after 3 years.

Question 2

[3 marks]

A machine valued at

\pounds 1200

loses

8\%

of its value each year. Find its value after 4 years, to the nearest penny.

Question 3

[3 marks]

Given

u_0 = 5

and

u_{n+1} = 2u_n - 3

, calculate

u_1, u_2, u_3

.

Question 4

[3 marks]

Given

u_0 = 4

and

u_{n+1} = \dfrac{1}{2}u_n + 6

, calculate

u_1, u_2, u_3

.

Question 5

[4 marks]

The first three terms of a linear recurrence relation

u_{n+1} = au_n + b

are

6, 10, 18

. Find

a

and

b

.

Question 6

[3 marks]

The recurrence

u_{n+1} = 0.75u_n + 12

approaches a limiting value

L

. Find

L

.

Section B

Topic practice

Build individual techniques

Question 7

[5 marks]

A savings account contains

\pounds 18000

. Interest of

3\%

is added each year before

\pounds 1000

is withdrawn. Write down a recurrence relation for the balance

u_n

, and find the balance after 3 years.

Question 8

[4 marks]

A patient is given

15

mg of a drug at the end of every day. During each day

40\%

of the drug already present is removed. Initially there is no drug in the bloodstream. Form a recurrence relation and calculate the amount immediately after the third dose.

Question 9

[5 marks]

A loan of

\pounds 4000

is charged interest at

2\%

per month. A repayment of

\pounds 350

is made at the end of every month. Write down a recurrence and find the amount outstanding after 3 payments.

Question 10

[4 marks]

A wildlife reserve contains

700

rabbits. Each month

30\%

of those present leave and

120

new rabbits arrive. Write down a recurrence relation and find its limiting population.

Question 11

[4 marks]

Given

u_0 = 10

and

u_{n+1} = 0.4u_n + 9

, calculate

u_1, u_2

and find the limiting value.

Question 12

[4 marks]

Three consecutive terms of a linear recurrence

u_{n+1} = au_n + b

are

4, 7, 13

. Determine

a

and

b

.

Question 13

[3 marks]

The recurrence

u_{n+1} = ku_n + 8

has a limiting value of

40

. Find

k

.

Question 14

[4 marks]

A recurrence is defined by

u_0 = 1000

and

u_{n+1} = 0.9u_n + 50

. Find the limiting value. State whether the first three terms are above or below it.

Section C

Exam-style and challenge

Combine skills, exam-style

Question 15

[4 marks]

The first three terms of a linear recurrence are

3, 8, 18

. Find the recurrence in the form

u_{n+1} = au_n + b

.

Question 16

[5 marks]

A club has

200

members. At the start of each year

75\%

renew their membership and

60

new people join. Model the total just after enrolment with a recurrence relation and determine whether the membership will ever reach

250

, assuming these rates remain constant.

Question 17

[5 marks]

Given

u_0 = 75

and

u_{n+1} = 0.6u_n + 18

, find the first value of

n

for which

u_n < 50

.

Question 18

[6 marks]

A debt of

\pounds 1200

attracts

5\%

interest monthly. A payment of

\pounds 200

is made after the interest is added. Find the number of monthly payments needed to clear the debt, assuming a smaller final payment is allowed.

Looking for the answers?

We do not publish the worked answers online. If you get stuck, bring the worksheet to a session and one of our Higher Maths tutors will go through the method with you, line by line, the way the SQA marking instructions award it.

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Frequently asked questions

Are the answers to these worksheets available online?

No. We keep the worked answers off the website on purpose. Pupils learn far more by attempting every question, checking against their notes and then going over the method with a tutor. Our students work through the full worked solutions in class, and if you are stuck on a particular question you can book a session and we will go through it with you step by step.

Are these official SQA or Qualifications Scotland papers?

No. Every question is newly written by Excel in MathSci to match the Higher Maths course specification. They are original practice resources, not official Qualifications Scotland papers, so use them alongside the real past papers rather than instead of them.

How should I use the three sections?

Section A checks the prerequisite skills you need before the topic makes sense, so start there if the topic feels shaky. Section B builds each technique on its own. Section C is exam-style and combines several skills, so treat it as a timed test once Sections A and B feel comfortable.

Can I print the worksheet?

Yes. Download the free PDF, which has space to write your working under every question, or print this page directly. The worksheet is free for personal study; please ask before reproducing it for a class or a commercial service.

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