Geometric progression

A geometric progression (GP) is a sequence of the form $a, ar, ar^2, ar^3, \ldots$, where $a$ is the first term and $r$ is the common ratio (with $r \neq 0$). Each term is obtained by multiplying the previous term by $r$.

Important properties

• The $n$th term of a GP is $a_n = a r^{n-1}$.

• The ratio between any two consecutive terms is always $r$.

• If $|r| < 1$, the terms get smaller; if $|r| > 1$, the terms get larger.

• The sum of the first $n$ terms is $S_n = a \frac{1 - r^n}{1 - r}$ for $r \neq 1$.