Sequences and the nth Term

GCSE Maths · Algebra

What a sequence is

A sequence is a list of numbers (terms) that follow a rule. Common types at GCSE are arithmetic (linear), geometric, and special sequences like square and triangular numbers.

Arithmetic (linear) sequences

Each term changes by a fixed amount — the common difference (d).

Example: 3, 7, 11, 15, … has a common difference of +4.

The nth term of a linear sequence

The nth term is a formula that lets you find any term. For a linear sequence it has the form dn + (something), where d is the common difference.

Steps to find it:

1. Find the common difference d — this is the number in front of n.

2. Work out the "zero term" (the term before the first) by subtracting d from the first term — this is the constant.

Example: find the nth term of 3, 7, 11, 15, …

Common difference = 4, so the rule starts 4n.
4×1 = 4, but the first term is 3, so subtract 1: nth term = 4n − 1.
Check: n = 3 → 4(3) − 1 = 11 ✓

Using the nth term

Find a term: substitute the term number for n. The 10th term of 4n − 1 = 4(10) − 1 = 39.
Is a number in the sequence? Set the nth term equal to it and solve; if n is a whole number, yes. Is 50 in 4n − 1? → 4n − 1 = 50 → 4n = 51 → n = 12.75, not whole → no.

Geometric sequences

Each term is multiplied by a fixed number (common ratio), e.g. 2, 6, 18, 54 (×3 each time).

Special sequences

Square numbers: 1, 4, 9, 16, 25 (n²).
Cube numbers: 1, 8, 27, 64 (n³).
Triangular numbers: 1, 3, 6, 10, 15 (add 2, then 3, then 4…).
Fibonacci: each term is the sum of the two before (1, 1, 2, 3, 5, 8…).

Worked example

Find the nth term of 5, 8, 11, 14, … and the 20th term.

Common difference = 3 → starts 3n. 3×1 = 3, first term is 5, so add 2 → nth term = 3n + 2.
20th term = 3(20) + 2 = 62. ✓

Common mistakes

Using the first term as the constant instead of the zero term.
Sign errors — check whether you add or subtract the constant using the first term.
Confusing arithmetic (add a fixed amount) with geometric (multiply by a fixed amount).

Exam tips

Always check your nth-term rule by substituting n = 1, 2, 3.
For "is X in the sequence?", solve for n and check it's a positive whole number.
Learn to recognise square, cube, triangular and Fibonacci sequences.

Key facts to remember

Arithmetic sequence: fixed common difference d; nth term = dn + (first term − d).
Substitute n to find any term; solve nth term = value to test membership (n must be a whole number).
Recognise square (n²), cube (n³), triangular and Fibonacci sequences.