Geometric series (adding the terms)

The sum of a geometric series

• To add the first $n$ terms of a geometric sequence, use:

$S_n = \frac{a\,(r^{\,n} - 1)}{r - 1} \qquad (r \neq 1)$

• $a$ — the first term.
• $r$ — the common ratio.
• $n$ — the number of terms.
• If $r$ is less than 1, it is tidier to use the equivalent $S_n = \dfrac{a\,(1 - r^{\,n})}{1 - r}$ so the top stays positive.

Using the formula

• Identify $a$, $r$ and $n$.
• Substitute carefully — the power $r^{\,n}$ uses $n$ (the number of terms), not $n - 1$.
• Work the power first, then the rest of the fraction.