What is a periodic sequence?

One whose terms repeat in a cycle: u{n+k}=un.

What is the order of a periodic sequence?

The number of terms in one cycle.

u1=2, u{n+1}=\frac{1}{1-un}: order?

3 (the terms 2,-1,\frac12 repeat).

How do you find constants in u{n+1}=pun+q?

Substitute known consecutive terms to form simultaneous equations.

Sequences and recurrence relationsEdexcel International A Level Maths: Flashcards

Card 1 of 120 of 12 known

Question

What is a recurrence relation?

Tap or press Space to reveal

Tap card or press Space to flip

See all 12 cards

What is a recurrence relation?: A rule giving $u_{n+1}$ in terms of $u_n$ , with a starting value.
What is an increasing sequence?: One with $u_{n+1}>u_n$ for all $n$ .
What is a decreasing sequence?: One with $u_{n+1}<u_n$ for all $n$ .
What is a periodic sequence?: One whose terms repeat in a cycle: $u_{n+k}=u_n$ .
What is the order of a periodic sequence?: The number of terms in one cycle.
How do you test if a sequence is increasing?: Show $u_{n+1}-u_n>0$ for every $n$ .
$u_1=3$ , $u_{n+1}=2u_n-1$ . Find $u_3$ .: $u_2=5$ , $u_3=9$ .
$u_n=n^2-6n+10$ . Find $u_{n+1}-u_n$ .: $2n-5$
What is $(-1)^n$ for odd $n$ and for even $n$ ?: $-1$ for odd $n$ , $+1$ for even $n$ .
$u_1=2$ , $u_{n+1}=\frac{1}{1-u_n}$ : order?: $3$ (the terms $2,-1,\frac12$ repeat).
How do you find $u_{50}$ in a periodic sequence of order $4$ ?: $50=4\times12+2$ , so $u_{50}=u_2$ .
How do you find constants in $u_{n+1}=pu_n+q$ ?: Substitute known consecutive terms to form simultaneous equations.

Exam questions on Sequences and recurrence relations

A sequence is given by $u_n=n^2-6n+10$ for $n\ge1$ .
Find an expression for $u_{n+1}-u_n$ in terms of $n$ , and hence state the values of $n$ for which $u_{n+1}>u_n$ .2 marks
A sequence is defined by $u_1=2$ and $u_{n+1}=\dfrac{1}{1-u_n}$ for $n\ge1$ .
Find the value of $u_{100}$ .2 marks
A sequence has $n$ th term $u_n=a+b(-1)^n$ , where $a$ and $b$ are constants, $u_1=3$ and $u_2=9$ .
Find the values of $a$ and $b$ .3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).