In an arithmetic sequence you add the same amount each step. A geometric sequence does the same trick with multiplication: you multiply by the same number — the common ratio r — to get from one term to the next.

Starting from a first term a, the terms are

Each term carries one more factor of r than the last, so the nth term is

The power is n-1, not n, because the first term a has been multiplied by r zero times — the index laws let us write that build-up of factors as a single power.

Take a = 3 and r = 2:

The 4th term is a\,r^{\,3} = 3 \times 2^3 = 24 — no need to step through all of them. With r > 1 the terms grow fast; with 0 < r < 1 each term is a fraction of the last, so they shrink toward zero.

Adding the terms: a geometric series

A series is the sum of the terms. Adding the first n terms of a geometric sequence gives

There is a neat closed form (multiply S_n by r, subtract, and almost everything cancels):

For 3, 6, 12, 24 with a = 3, r = 2, n = 4: S_4 = \dfrac{3(1 - 2^4)}{1 - 2} = \dfrac{3 \times (-15)}{-1} = 45, and indeed 3 + 6 + 12 + 24 = 45.

The sum to infinity

Something special happens when |r| < 1. Each term is smaller than the last, so the running total piles up less and less — and the partial sums S_n creep toward a fixed value rather than running away. In the formula, r^{\,n} \to 0 as n \to \infty, leaving the sum to infinity:

The classic case is a = 1, r = \tfrac{1}{2}:

Watch the partial sums below: each new term adds half of the gap that remains, so the total edges ever closer to 2 without ever passing it. (If |r| \ge 1 the terms do not shrink, the total never settles, and there is no sum to infinity.)