Geometric sequences and series

Sequences with a constant common ratio — the nth term, the sum of n terms, and the sum to infinity (and exactly when it exists).

Pure · Sequences & series

Sequences with a constant common ratio — the nth term, the sum of n terms, and the sum to infinity (and exactly when it exists).

Why it works

A geometric sequence multiplies by the same number each step — the common ratio

r

a,\ ar,\ ar^2,\ ar^3,\ \dots

By the

n

th term you've multiplied by

r

a total of

(n - 1)

times, so

u_n = ar^{\,n-1}.

(Find

r

by dividing any term by the one before it:

r = \dfrac{u_2}{u_1}

.)

Sum of $n$ terms. A clever trick: write

S_n = a + ar + \dots + ar^{n-1}

, multiply by

r

, and subtract — almost everything cancels, leaving

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

Sum to infinity. If

|r| < 1

, then

r^n \to 0

n \to \infty

, so the sum settles on a finite value:

S_\infty = \frac{a}{1 - r}, \qquad |r| < 1.

This only works when

|r| < 1

— if

|r| \ge 1

the terms don't shrink and the sum runs off to infinity (no sum to infinity exists).

Watch for the two classic slips: using

ar^n

instead of

ar^{n-1}

, and trying to use

S_\infty

when

|r| \ge 1