mathsGrade 11-122 min read

Arithmetic and geometric series on one page

Two kinds of sequence cover most of what exams ask: ones that grow by adding, and ones that grow by multiplying. Here are the formulas, the one that surprises everyone, and how to tell them apart.

A sequence is a list of numbers; a series is what you get when you add them up. Almost everything at this level is one of two types, and the first job is always to spot which.

Telling them apart

Arithmetic — each term is the one before plus a fixed number, the common difference $d$ . ( $3, 7, 11, 15, \dots$ , with $d = 4$ .)
Geometric — each term is the one before times a fixed number, the common ratio $r$ . ( $3, 6, 12, 24, \dots$ , with $r = 2$ .)

Add to get the next term, it's arithmetic. Multiply, it's geometric. Check the gap between terms first.

Arithmetic formulas

The $n$ -th term, and the sum of the first $n$ terms:

u_n = a + (n-1)d, \qquad S_n = \frac{n}{2}\big(2a + (n-1)d\big)

where $a$ is the first term. The sum formula is just "number of terms times the average of the first and last."

Geometric formulas

u_n = a\,r^{\,n-1}, \qquad S_n = \frac{a\,(1 - r^{\,n})}{1 - r}

The result that surprises everyone

If a geometric series shrinks — that is, if $\lvert r\rvert$ is less than $1$ — then adding up infinitely many terms gives a finite answer:

S_\infty = \frac{a}{1 - r} \qquad (\lvert r\rvert < 1)

Half plus a quarter plus an eighth plus... forever, never passes $1$ . Each term is so much smaller than the last that the total settles on a fixed number. This only works when the terms shrink; a growing series ( $\lvert r \rvert \ge 1$ ) runs off to infinity.

Identify the type, then list $a$ , $d$ or $r$ , and $n$ before reaching for a formula. Picking the geometric formula for an arithmetic sequence is the commonest error — and it's avoided entirely by checking "add or multiply?" first.

Last revised 20 November 2025.