geometric sequence

A geometric sequence grows by multiplying by the same fixed number every step, rather than adding. Think of money in an account that grows by 10% a year, or a bacterial colony that doubles each hour: 1, 2, 4, 8, 16, ... — each term is twice the one before.

That fixed multiplier is the common ratio r. Multiplying the first term by r repeatedly gives the explicit formula a_n = a_1 · r^(n−1). Because the position n sits in the exponent, geometric sequences are tied to exponential growth and decay rather than to straight lines.

The defining test is that consecutive ratios are all equal: a_2 / a_1 = a_3 / a_2 = r. The ratio can be a fraction (then the terms shrink toward 0) or negative (then the signs alternate). The first term should not be 0, since you cannot recover a ratio from a term of 0.