Geometric Series Formula

The sum of a geometric sequence $a, ar, ar^2, ..., ar^{n-1}$ equals $S = a \times \frac{r^n - 1}{r - 1}$ when $r \neq 1$.

For example, $1 + 2 + 4 + 8 + 16$ has $a = 1$, $r = 2$, $n = 5$. So $S = 1 \times \frac{2^5 - 1}{2 - 1} = \frac{32 - 1}{1} = 31$.

When $r = 1$, all terms are the same, so the sum is just $a \times n$. The formula breaks down because you'd divide by zero.