Sequences (Edexcel IGCSE Maths): Revision notes

Section 1

What is a sequence, and how do term-to-term rules work?

A sequence is an ordered list of numbers called terms, e.g. $3, 7, 11, 15, \dots$

A term-to-term rule tells you how to get from one term to the next. For example: "add 4 to the previous term".

To use a term-to-term rule you need to know the first term and the rule itself.
Term-to-term rules are quick to state but slow to use if you want, say, the 100th term — you would have to work through every term before it.
Common types: arithmetic (add/subtract a fixed amount), geometric (multiply/divide by a fixed amount), and rules involving previous two terms (e.g. Fibonacci-style: add the two previous terms).

Key termssequencetermterm-to-term rule

Example

Sequence: $2, 5, 8, 11, \dots$ Rule: "start at 2, add 3 each time". Next term after 11 is $11+3=14$ .

Common mistake

Don't confuse the term-to-term rule with the position-to-term rule — a term-to-term rule alone cannot give you the 50th term directly.

Section 2

How do I find the nth term of a linear sequence?

A linear sequence has a constant difference between consecutive terms — this constant is called $d$ .

The position-to-term rule (nth term formula) for a linear sequence is: $u_n = dn + (a - d)$ where $d$ is the common difference and $a$ is the first term.

Method:

Find the common difference $d$ (subtract consecutive terms).
The nth term starts with $dn$ .
Compare $dn$ to the actual sequence to find the number you add or subtract.

Once you have $u_n$ , you can substitute any position number $n$ to get that term directly — no need to list every term before it.

Key termslinear sequencecommon differencenth termposition-to-term rule

Example

Sequence: $5, 8, 11, 14, \dots$ Common difference $d=3$ , so nth term starts $3n$ . When $n=1$ : $3(1)=3$ , but the term is 5, so add 2. nth term $=3n+2$ . Check $n=4$ : $3(4)+2=14$ . Correct.

Exam tip

The coefficient of $n$ in the nth term is always equal to the common difference $d$ .

Section 3

How do I find the nth term of a quadratic sequence?

A quadratic sequence has an nth term of the form $an^2+bn+c$ . You can spot one because the first differences (between consecutive terms) are not constant, but the second differences (differences of the differences) are constant.

Method:

Find the first differences, then the second differences.
The second difference equals $2a$ , so $a = \dfrac{\text{second difference}}{2}$ .
Subtract $an^2$ from each term to leave a linear sequence in $n$ .
Find the nth term of that linear sequence — this gives $bn+c$ .
Combine: nth term $= an^2+bn+c$ .

Key termsquadratic sequencefirst differencesecond difference

Example

Sequence: $2, 7, 14, 23, 34$ . First differences: $5, 7, 9, 11$ . Second differences: $2, 2, 2$ (constant), so it's quadratic with $a=1$ . Subtract $n^2$ : $2-1=1$ , $7-4=3$ , $14-9=5$ , $23-16=7$ — this is linear with nth term $2n-1$ . So overall nth term $=n^2+2n-1$ .

Common mistake

A very common error is forgetting to divide the second difference by 2 to get $a$ — students use the second difference itself as the coefficient of $n^2$ .

Section 4

What are special sequences I should recognise?

Some sequence types come up repeatedly and are worth recognising instantly:

Square numbers: $1, 4, 9, 16, 25, \dots$ ( $n^2$ )
Cube numbers: $1, 8, 27, 64, \dots$ ( $n^3$ )
Triangular numbers: $1, 3, 6, 10, 15, \dots$ (nth term $=\dfrac{n(n+1)}{2}$ )
Fibonacci-type sequences: each term is the sum of the two previous terms, e.g. $1, 1, 2, 3, 5, 8, \dots$
Powers of a number: e.g. powers of 2: $2, 4, 8, 16, \dots$

Recognising these instantly saves time — you may be asked to identify the type or continue the pattern without deriving a full formula.

Key termssquare numberscube numberstriangular numbersFibonacci-type sequence

Think of it like this

Think of triangular numbers as stacking rows of dots to build a triangle — row $n$ adds $n$ more dots than the row before, so the total is a running sum.

Section 5

How do geometric progressions work?

A geometric progression (GP) has a constant common ratio $r$ between consecutive terms (multiply, don't add).

$u_n = ar^{n-1}$ where $a$ is the first term and $r$ is the common ratio.

Find $r$ by dividing any term by the term before it: $r = \dfrac{u_{n+1}}{u_n}$ .
$r$ can be a fraction (sequence decreasing towards zero) or negative (terms alternate sign).
Unlike linear/quadratic sequences, you cannot find $r$ by subtracting terms — you must divide.

Key termsgeometric progressioncommon ratio

Example

Sequence: $3, 6, 12, 24, \dots$ Here $a=3$ , $r=2$ , so $u_n = 3(2)^{n-1}$ . The 6th term is $3(2)^5 = 96$ .

Common mistake

Don't try to subtract terms to find $r$ in a geometric progression — always divide consecutive terms instead.

Must Know

Linear sequence nth term: $u_n = dn + (a-d)$ , where $d$ is the common difference.
Quadratic sequence: constant second difference; coefficient of $n^2$ is half the second difference.
Square numbers ( $n^2$ ), cube numbers ( $n^3$ ), and triangular numbers ( $\frac{n(n+1)}{2}$ ) should be recognised on sight.
Geometric progression nth term: $u_n = ar^{n-1}$ ; find $r$ by dividing, not subtracting.
Always check your nth term formula by substituting $n=1, 2, 3$ back into it.
A term-to-term rule generates terms one at a time; a position-to-term (nth term) rule jumps straight to any term.

That's the notes covered.

Carry on to the next subtopic.