Sequences Notes

Edexcel IGCSE Maths: Revision notes

Key facts

A term-to-term rule gives the next term from the previous one; an nth term rule jumps to any term.
Linear sequence: $u_n=dn+(a-d)$ , where $d$ is the common difference and $a$ the first term.
Quadratic sequence: constant second difference; the coefficient of $n^2$ is half of it.
Recognise square, cube, triangular, Fibonacci-type and powers of 2 on sight.
Geometric progression: $u_n=ar^{n-1}$ , with $r$ found by dividing.

Term-to-term rules

A sequence is an ordered list of terms; a term-to-term rule says how to get from one term to the next.

A sequence is an ordered list of numbers called terms. A term-to-term rule such as "add 4 to the previous term" needs the first term and the rule.

It is slow for the 100th term. Common types are arithmetic (add a fixed amount), geometric (multiply by a fixed amount) and rules using the previous two terms (Fibonacci-style).

1
2
first term
2
5
add 3
3
8
add 3
4
11
add 3
5
14
add 3

Rule: start at 2, add 3 each time

A sequence starts at 4 with the rule "multiply by 2, then subtract 1". What are the next three terms?

Linear nth term

Use the common difference as the coefficient of n, then adjust for the first term.

A linear sequence has a constant difference $d$ . Its nth term (position-to-term rule) is $u_n=dn+(a-d)$ , so you can find any term directly.

The terms lie on a straight line: 5, 8, 11, 14 fit $u=3n+2$ .

The terms 5, 8, 11, 14 lie on the straight line u = 3n + 2

Worked example

Find the nth term of $5,8,11,14,\dots$

1
Common difference $8-5=3$ , so start with $3n$ .
2
At $n=1$ : $3(1)=3$ but the term is 5, so add 2.
3
Check $n=4$ : $3(4)+2=14$ .

Find the nth term of $7,11,15,19,\dots$

Quadratic nth term

If the second differences are constant, halve them to get the coefficient of n².

A quadratic sequence has nth term $an^2+bn+c$ . The first differences change but the second differences are constant, and $a=\dfrac{\text{second difference}}{2}$ .

Subtract $an^2$ from each term to leave a linear sequence, find its nth term, then combine.

Quadratic: 2, 7, 14, 23, 34 Linear: 5, 8, 11, 14 First differences 5, 7, 9, 11 3, 3, 3 (constant) Second differences 2, 2, 2 (constant) 0, 0, 0

	Quadratic: 2, 7, 14, 23, 34	Linear: 5, 8, 11, 14
First differences	5, 7, 9, 11	3, 3, 3 (constant)
Second differences	2, 2, 2 (constant)	0, 0, 0

Quadratic: 2, 7, 14, 23, 34

First differences:: 5, 7, 9, 11
Second differences:: 2, 2, 2 (constant)

Linear: 5, 8, 11, 14

First differences:: 3, 3, 3 (constant)
Second differences:: 0, 0, 0

1
Differences
find first and second differences
2
Find a
second difference ÷ 2
3
Subtract an²
from each term
4
Linear nth term
gives bn + c
5
Combine
an² + bn + c

Quadratic nth term

Worked example

Find the nth term of $2,7,14,23,34,\dots$

1
First differences $5,7,9,11$ ; second differences all 2, so $a=1$ .
2
Subtract $n^2$ : $1,3,5,7$ , which is $2n-1$ .

A sequence $3,9,19,33,\dots$ has constant second difference 4. What is its nth term?

Special sequences

Spot these on sight to save time.

You may be asked to identify the type or continue the pattern without deriving a formula.

n^2

	Terms	nth term or rule
Square numbers	1, 4, 9, 16, 25	$n^2$
Cube numbers	1, 8, 27, 64	$n^3$
Triangular numbers	1, 3, 6, 10, 15	$\dfrac{n(n+1)}{2}$
Fibonacci-type	1, 1, 2, 3, 5, 8	Add the two previous terms
Powers of 2	2, 4, 8, 16	$2^n$

Terms

Square numbers:: 1, 4, 9, 16, 25
Cube numbers:: 1, 8, 27, 64
Triangular numbers:: 1, 3, 6, 10, 15
Fibonacci-type:: 1, 1, 2, 3, 5, 8
Powers of 2:: 2, 4, 8, 16

nth term or rule

Square numbers:: $n^2$
Cube numbers:: $n^3$
Triangular numbers:: $\dfrac{n(n+1)}{2}$
Fibonacci-type:: Add the two previous terms
Powers of 2:: $2^n$

What is the next triangular number after 15?

Geometric progressions

A geometric progression multiplies by a constant ratio each time.

A geometric progression has a constant common ratio $r$ , and $u_n=ar^{n-1}$ with first term $a$ .

Find $r$ by dividing a term by the one before it, not by subtracting. $r$ can be a fraction (decreasing) or negative (alternating signs).

The geometric progression 3, 6, 12, 24, 48, 96 (r = 2)

Worked example

Find the 6th term of $3,6,12,24,\dots$

1
$a=3$ and $r=6\div3=2$ , so $u_n=3(2)^{n-1}$ .
2
$u_6=3(2)^5=3\times32$ .

What is the common ratio of $81,27,9,3,\dots$ ?

Try an exam question

Find an expression, in terms of $n$ , for the nth term of the sequence $3,8,15,24,\dots$

[3 marks]

That's the notes covered.

Carry on to the next subtopic.