geometric series

A geometric series is the sum of the terms of a geometric sequence — adding up a list where each term is a fixed multiple of the one before. For example 1 + 2 + 4 + 8 + 16 adds the first terms of a doubling sequence.

There is a clean closed-form for the sum of the first n terms: S_n = a_1 · (1 − r^n) / (1 − r), valid whenever r ≠ 1. The trick behind it is to multiply the whole sum by r and subtract, which cancels almost every term. (If r = 1 every term equals a_1, so the sum is simply n · a_1.)

This formula spares you adding the terms one by one and underlies everything from loan payments to the sum of an infinite geometric series when |r| < 1. Watch the exponent: it is r^n, not r^(n−1), even though the n-th term itself uses r^(n−1).