How to find the nth term of a linear sequence

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

A sequence is linear (arithmetic) if the difference between every pair of consecutive terms is the same. That common difference, d, is the coefficient of n in the nth term; the constant is what you would get at "term zero", one step before the first term. Once you have the rule, any term — the 50th, the 100th — is a single substitution away. Here is how it works, step by step.

The rule

nth term = dn + c, where d is the common difference and c = first term − d

In a linear sequence every step forward adds exactly the same difference d — here d = 4.

Worked examples, step by step

Checking a number sequence

Is the sequence 5, 9, 13, 17 linear? Find its nth term.

Work out the differences between consecutive terms: 9 − 5 = 4, 13 − 9 = 4, 17 − 13 = 4
All the differences are 4, so the sequence is linear with d = 4
The constant is the first term minus d: 5 − 4 = 1
Conclusion: the nth term is 4n + 1 (check: n = 4 gives 4 × 4 + 1 = 17)

Using the nth term to find a far-away term

The nth term of a sequence is 3n + 2. Find the 50th term, and show that consecutive terms always differ by 3.

Substitute n = 50: 3 × 50 + 2 = 152
Write the (n + 1)th term: 3(n + 1) + 2 = 3n + 5
Subtract: (3n + 5) − (3n + 2) = 3
The difference 3 is constant and does not depend on n — that is exactly what makes the sequence linear

A counter-example — a sequence that is not linear

Is the sequence 1, 2, 4, 8 linear?

Check the differences: 2 − 1 = 1, but 4 − 2 = 2
One different difference is enough to rule it out — the sequence is not linear
By the way: the ratio between consecutive terms is constant (2), so this is a geometric sequence

Finding the sequence from two given terms

In a linear sequence the 3rd term is 11 and the 7th term is 23. Find d and the first term.

Between the 3rd and the 7th term there are 4 "steps" of d, so 23 − 11 = 4d
So 12 = 4d and d = 3
Go back two steps from the 3rd term: first term = 11 − 2 × 3 = 5
Check: the sequence 5, 8, 11, 14, 17, 20, 23 — and the 7th term is indeed 23

Now try it yourself

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Formulae sheet

The nth term and the other sequence facts, with what is given in the exam and what is not.

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Learn sequences

From spotting the pattern to GCSE questions — explanation and guided practice.

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Geometric sequences

The other side of the coin — constant ratio instead of constant difference.

Frequently asked questions

Is it enough to check the first three terms?

For an exam question that gives you the first few terms, yes — find the common difference and write the rule. But equal differences at the start do not prove a sequence continues that way for ever; a full proof works out the (n + 1)th term minus the nth term from the rule and shows the result does not depend on n.

What is the difference between a linear sequence and a geometric sequence?

In a linear sequence the difference between consecutive terms is constant (you add the same number each time); in a geometric sequence the ratio is constant (you multiply by the same number each time). 5, 8, 11 is linear; 5, 10, 20 is geometric.

How do you decide whether a number is in a sequence?

Set the nth term equal to the number and solve for n. If n is a positive whole number the number is in the sequence. For 4n + 1 = 101, n = 25, so 101 is the 25th term; for 4n + 1 = 50, n is not a whole number, so 50 is not in the sequence.

How do you add up the terms of a linear sequence?

Sum = number of terms × (first term + last term) ÷ 2 — the count times the mean of the first and last terms. For 1 to 100: 100 × 101 ÷ 2 = 5050. The sum formula itself is A level rather than GCSE, but the trick is useful.

✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specificationsLast updated: 13.09.2026