What is a periodic sequence?

One where u{n+k}=un for all n; the same k terms repeat.

What is the order of a periodic sequence?

The number of terms in one cycle.

How do you sum many terms of a periodic sequence?

Sum one cycle, multiply by the number of whole cycles, add the leftover terms.

an=\frac{n}{n+2}: is it increasing?

Yes, a{n+1}-an=\frac{2}{(n+2)(n+3)}0.

Sequences and recurrence relationsAQA A-Level Maths: Flashcards

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What is a recurrence relation?

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What is a recurrence relation?: A rule $x_{n+1}=f(x_n)$ giving each term from the previous one, with a starting value.
$u_1=3$ , $u_{n+1}=2u_n-1$ : first five terms?: $3,5,9,17,33$
When is a sequence increasing?: $u_{n+1}>u_n$ for all $n$ .
When is a sequence decreasing?: $u_{n+1}<u_n$ for all $n$ .
How do you prove a sequence is increasing?: Show $u_{n+1}-u_n>0$ for all $n$ .
What is a periodic sequence?: One where $u_{n+k}=u_n$ for all $n$ ; the same $k$ terms repeat.
What is the order of a periodic sequence?: The number of terms in one cycle.
$x_1=2$ , $x_{n+1}=\frac{1}{1-x_n}$ : its terms and order?: $2,-1,\frac12,2,\dots$ , order $3$
How do you find $x_{100}$ in a sequence of order $3$ ?: $100=3\times33+1$ , so $x_{100}=x_1$ .
How do you sum many terms of a periodic sequence?: Sum one cycle, multiply by the number of whole cycles, add the leftover terms.
Check that $u_n=2^n+1$ fits $u_{n+1}=2u_n-1$ .: $2(2^n+1)-1=2^{n+1}+1=u_{n+1}$
$a_n=\frac{n}{n+2}$ : is it increasing?: Yes, $a_{n+1}-a_n=\frac{2}{(n+2)(n+3)}>0$ .

Exam questions on Sequences and recurrence relations

A sequence is defined by $u_1=3$ and $u_{n+1}=2u_n-1$ for $n\ge1$ .
Show that $u_n=2^n+1$ satisfies both $u_1=3$ and the relation $u_{n+1}=2u_n-1$ .2 marks
A sequence has $n$ th term $a_n=\frac{n}{n+2}$ for $n\ge1$ .
Prove that the sequence is increasing.2 marks
A sequence is defined by $x_1=2$ and $x_{n+1}=\frac{1}{1-x_n}$ for $n\ge1$ .
Find $x_2$ , $x_3$ and $x_4$ , and state what this shows about the sequence.3 marks

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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).