geometric sequence

A geometric sequence is a sequence where a subsequent element is the previous element multiplied by a constant value called the common ratio.

Formally,

g_{n + 1} = g_{n} \cdot r

g_{n} = g_{k} \cdot r^{n - k}

The partial sum of a geometric sequence is

\sum_{k = 0}^{n} a \cdot r^{k} = \frac{a (1 - r^{n + 1})}{1 - r}

\sum_{k = 0}^{\infty} a \cdot r^{k} = \frac{a}{1 - r}

when $| r | < 1$ , and diverges otherwise.