Mathematics

What Is an Arithmetic Sequence?

Each term adds the same number. How to test for one, why a + (n − 1)d and the British 3n + 2 are the same rule, and how to reach the 20th term directly.

Arithmetic sequence

A sequence in which each term is found by adding the same fixed number, called the common difference, to the term before it.

Also called: Linear sequence, Arithmetic progression
Where students meet it: Grade 8 mathematics, when nth-term rules are first written, and on every GCSE and IGCSE paper as well as the Digital SAT.

The short answer
An arithmetic sequence adds the same fixed number each time, called the common difference. In 5, 8, 11, 14 that number is 3. The nth term is a + (n − 1)d, so the 20th term is 5 + 19 × 3 = 62. British papers usually write the same rule more compactly as 3n + 2.

An example

5, 8, 11, 14, … d = 3 nth term = 3n + 2 20th term = 62

The gap between consecutive terms is 3 every time, so the sequence is arithmetic and the common difference is 3. Because the terms go up in threes, the rule must be built on 3n: the multiples of 3 are 3, 6, 9, 12, and the sequence sits two above each of them, so the rule is 3n + 2. Substituting n = 20 gives 62 directly, without listing the intervening fifteen terms — which is the entire reason for having a rule.

Test the differences before anything else

Subtract each term from the one after it. If every answer is the same, the sequence is arithmetic and that repeated answer is the common difference. If they are not the same, no amount of staring at the numbers will make it arithmetic and you need a different tool.

Three sequences make the boundary clear. 5, 8, 11, 14 gives differences of 3, 3, 3 — arithmetic. 3, 6, 12, 24 gives 3, 6, 12, which are not equal, and this sequence multiplies rather than adds, making it geometric. 1, 4, 9, 16 gives 3, 5, 7 — also not constant, but those differences increase steadily, which signals a quadratic sequence.

The common difference can be negative, and nothing about the method changes. 20, 17, 14, 11 has d = −3, and the terms fall rather than rise.

Two ways of writing one rule

The international form is a + (n − 1)d, where a is the first term and d the common difference. For 5, 8, 11, 14 that gives 5 + 3(n − 1). Expand the bracket: 5 + 3n − 3, which tidies to 3n + 2 — the answer a British GCSE paper expects when it asks for the nth term.

So the two are not competing methods but the same expression before and after simplifying. It is worth doing that expansion once, because students who have met both often assume one of them must be wrong.

The compact form also tells you something at a glance. The number in front of n is always the common difference, and the constant is what the zeroth term would have been — the value one step before the sequence started. For 3n + 2, stepping back from 5 by 3 gives 2, which is exactly the constant.

Reaching a distant term, and going backwards

The forward direction is substitution. The 20th term of 3n + 2 is 3 × 20 + 2 = 62, and the 100th is 302, neither of which requires writing out the sequence.

The backwards direction is where the rule earns its keep. Asked whether 92 is in the sequence, set 3n + 2 = 92 and solve: n = 30. That is a whole number, so yes, 92 is the 30th term. Asked about 100, the same equation gives n = 32.67, which is not a term number at all, so 100 is not in the sequence. Being able to say which term a value is — or that it is none of them — is the standard second half of these questions.

Common questions

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

Find the common difference and put it in front of n. Then compare the multiples of that number with your sequence and adjust by a constant. For 5, 8, 11, the difference is 3, the multiples of 3 are 3, 6, 9, and the sequence is two higher throughout, so the rule is 3n + 2.

Is a linear sequence the same as an arithmetic sequence?

Yes. British schools tend to say linear sequence, because plotting the terms against their positions gives points in a straight line, while arithmetic sequence is the international and SAT phrasing. Arithmetic progression appears in older textbooks and means the same thing again.

Can the common difference be negative or a fraction?

Both. 20, 17, 14, 11 has a common difference of −3 and its nth term is 23 − 3n. A sequence rising by half each time has d = 0.5. The only requirement is that the same number is added every time, whatever kind of number it happens to be.

What if the differences are not constant?

Then the sequence is not arithmetic and you check two other things. Divide consecutive terms — if the answers match, it is geometric. Take the differences of the differences — if those are constant, it is quadratic and the nth term will contain an n² term. Squares and triangular numbers both fall into that second group.

Last updated August 21, 2026