Further Mathematics · Pearson Edexcel 9FM0 · Option FP2 · Topic 6

Further sequences and series

First- and second-order recurrence relations, solved in closed form and used in modelling.

3 sub-topics
Taught in Year 13
45 practice questions
3 animated lessons

Practice questions (45) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

6.1Recurrence relationsY13
6.2Solving recurrence relationsY13
6.3Proving closed forms by inductionY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 45 questions in this topic, free, no account needed.

Further sequences and series

Not needed for Year 12Easy

Forming a recurrence · Year 13 · 1 mark

A savings account earns 2% per year and £100 is added each year.

Write a recurrence for the balance uₙ.
(1)

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$u_{n+1} = 1.02u_n + 100$

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further sequences and series

Not needed for Year 12Medium

Solving a recurrence · Year 13 · 2 marks

Solve $u_{n+2} = 2u_{n+1} + 3u_n$ .
(2)

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$\lambda^2 - 2\lambda - 3$ ${}= 0$ ${}\Rightarrow \lambda$ ${}= -1, 3$
$u_n = A(-1)^n + B(3)^n$

Total2

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further sequences and series

Not needed for Year 12Hard

Forming a recurrence relation · Year 13 · 6 marks

A code is a string of letters from the set

\{A, B, C, D\}

. A code is valid if it never has two A's next to each other. Let

u_n

be the number of valid codes of length

n

.

(a)
Explain why $u_n = 3u_{n - 1} + 3u_{n - 2}$ for $n \geq 3$ .
(3)
(b)
Find $u_1$ and $u_2$ , and hence find $u_6$ .
(3)

Check your answer before you look at the solution

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Part (a)3 marks

Split by the last letter.A valid code ending in a letter other than $A$ : 3 choices for the last letter, after any valid code of length $n - 1$ .
A valid code ending in A: the letter before must not be A (3 choices), after any valid code of length n − 2.
$u_n = 3u_{n - 1} + 3u_{n - 2}$

Part (b)3 marks

Only AA is invalid at length 2. $u_1 = 4, \ u_2 = 16 - 1 = 15$
$u_3 = 57, \ u_4 = 216, \ u_5 = 819$
$u_6 = 3105$

Total6

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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