IGCSEMathsEdexcelMaths AHigherRevision NotesSequences, Functions & GraphsSequencesArithmetic Sequences

Arithmetic Sequences (Edexcel IGCSE Maths A: Higher): Revision Note

Exam code: 4MA1

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on 3 March 2026

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Arithmetic sequences

What is an arithmetic sequence?

An arithmetic sequence is a sequence where terms increase by the same amount each time
- The amount it increases by is called the common difference
  - For example: 3, 5, 7, 9, ...
  - The common difference is 2
Common differences can be negative
- These arithmetic sequences decrease by the same amount each time
  - For example: 11, 8, 5, 2, -1, ...
  - The common difference is -3
Arithmetic sequences are also called linear sequences

What is the formula for the nth term of an arithmetic sequence?

The formula for the $n$ ^th term of an arithmetic sequence is
$u_{n} = a + (n - 1) d$
- $a$ is the first term
- $d$ is the common difference
- $u_{n}$ is the $n$ ^th term
  - e.g. $n = 5$ gives the fifth term, $u_{5}$

How do I use the formula for the nth term of an arithmetic sequence?

You can substitute in the values of $a$ , $d$ and $n$ to find a particular term
- e.g. $a = 10$ , $d = 4$ and $n = 3$ gives $u_{3} = 10 + (3 - 1) \times 4 = 18$
  - So 18 is the 3^rd term
You can substitute in the values of $a$ and $d$ to find an expression for $u_{n}$
- e.g. $a = 10$ and $d = 4$ gives $u_{n} = 10 + (n - 1) \times 4$
  - This simplifies to $u_{n} = 10 + 4 (n - 1) = 10 + 4 n - 4 = 6 + 4 n$
You can form equations in terms of $a$ and $d$ if given information about terms
- e.g. If the 5^th term is 11,
  - $u_{5} = 11$ giving $a + (5 - 1) d = 11$
  - This simplifies to $a + 4 d = 11$
- Forming two different equations in $a$ and $d$ can lead to simultaneous equations

Examiner Tips and Tricks

You are not given the formula for the $n$ ^th term in the formula booklet.

Worked Example

The 6^th and 21^st terms in an arithmetic sequence are 13 and 43, respectively.

Find the first term, $a$ , and the common difference, $d$ .

Answer:

The formula for an arithmetic sequence is $u_{n} = a + (n - 1) d$
If the 6^th term is 13, then $u_{6} = 13$
Substitute $n = 6$ and $u_{6} = 13$ into the formula

$\begin{array}{rcl} u_{6} & = & a + (6 - 1) d \\ 13 & = & a + 5 d \end{array}$

Similarly, if the 21^st term is 43, then $u_{21} = 43$
Substitute $n = 21$ and $u_{21} = 43$ into the formula

$\begin{array}{rcl} u_{21} & = & a + (21 - 1) d \\ 43 & = & a + 20 d \end{array}$

These give two different equations in $a$ and $d$
Solve the equations simultaneously (for example, eliminate $a$ )

$- a + 20 d = 43 a + 5 d = 13 15 d = 30$

$d = \frac{30}{15} = 2$

Substitute $d = 2$ into the first equation and solve for $a$

$\begin{array}{rcl} 13 & = & a + 5 \times 2 \\ 13 & = & a + 10 \\ 3 & = & a \end{array}$

Write out the final answer

The first term is 3 and the common difference is 2

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Questions students ask about Arithmetic Sequences

Real questions from students revising this topic on Save My Exams

How do you find the common difference when the terms are algebraic expressions?

Subtract each term from the one after it, exactly as with numbers — and because the difference must be the same every time, you can set the two expressions equal and solve.

d = (2nd term) − (1st term) = (3rd term) − (2nd term)
Put brackets round the term you are subtracting so the signs are right
Solve the resulting equation for the unknown, then work out the actual terms

Example: x, 2x + 1 and 4x − 3 are the first three terms of an arithmetic sequence.

First difference: (2x + 1) − x = x + 1
Second difference: (4x − 3) − (2x + 1) = 2x − 4
Set equal: x + 1 = 2x − 4, so x = 5
The terms are 5, 11, 17
d = 11 − 5 = 6 (check: 17 − 11 = 6)

If there is no unknown to solve for, e.g. 3p, 3p + q, 3p + 2q, just subtract: d = q.

How do you find the first term of an arithmetic sequence?

The first term is a = u₁, so if it is not listed you substitute what you know into uₙ = a + (n − 1)d and solve for a.

If you know one term and d

e.g. the 5th term is 11 with d = 2: a + (5 − 1)d = 11, so a + 8 = 11, giving a = 3
You can also just count backwards, subtracting d each time

If you know two terms (the page's example)

6th term is 13, 21st term is 43
13 = a + 5d and 43 = a + 20d
Subtract: 15d = 30, so d = 2
Substitute: 13 = a + 10, so a = 3

If you have the nth term expression

Substitute n = 1, e.g. uₙ = 6 + 4n gives u₁ = 10

Remember uₙ = a + (n − 1)d — it is not in the formula booklet.

What does a represent in the nth term formula?

In uₙ = a + (n − 1)d, a is the first term of the sequence — the value when n = 1.

d is the common difference (how much it goes up or down by each time)
n is the position of the term, e.g. n = 5 for the 5th term
uₙ is the value of that term

Why it works: you start at a and add d a total of (n − 1) times to reach the nth term.

Page example: a = 10, d = 4

The sequence is 10, 14, 18, 22, ...
u₃ = 10 + (3 − 1) × 4 = 18

Watch out: once the formula is simplified, the number on its own is not a.

uₙ = 10 + (n − 1) × 4 simplifies to uₙ = 6 + 4n
The first term is still 10, not 6 (check: 6 + 4 × 1 = 10)

So read a from the a + (n − 1)d form, or find u₁ by substituting n = 1.