# How do you find the sum of the infinite geometric series 36+24+16+...?

Jun 7, 2016

$108$

#### Explanation:

In any geometric series, we need two pieces of information to find its sum to the $n$-th term, and thus the sum to infinity.

Firstly, we need the first term, $a$. This is obviously $36$ in this expression.

Secondly, we need the common ratio, $r$,. We find this by taking $\frac{24}{36} = \frac{2}{3}$ and confirm that this is the common ratio by taking $\frac{16}{24} = \frac{2}{3}$.

Thus, $a = 36$ and $r = \frac{2}{3}$. Also note that $\left\mid r \right\mid = \frac{2}{3} < 1$.

We then apply the formula for sum of geometric series to the $n$-th term when $\left\mid r \right\mid < 1$: ${S}_{n} = \frac{a \left(1 - {r}^{n}\right)}{1 - r} = \frac{36 \left(1 - {\left(\frac{2}{3}\right)}^{n}\right)}{1 - \left(\frac{2}{3}\right)} = 108 \left(1 - {\left(\frac{2}{3}\right)}^{n}\right)$

The sum to infinity, ${S}_{\infty}$ is defined to be the limit of ${S}_{n}$ as $n \to \infty$.

$n \to \infty$, ${\left(\frac{2}{3}\right)}^{n} \to 0$, ${S}_{n} \to 108 - 0 = 108$, ${S}_{\infty} = 108$.