# How do you find the sum of the infinite geometric series 1/4 + 1/8 + 1/16 + 1/32 + ..?

Nov 12, 2015

$\frac{1}{2}$

#### Explanation:

In a geometric series, we multiply by some number $r$ to get to the next term.

The trick is to find this number $r$. If $| r | < 1$ then you can use the following expression to find the sum:
$\frac{a}{1 - r}$, where $a$ is the first term of the series.

we know this:
$\frac{1}{4} \cdot r = \frac{1}{8}$
$r = \frac{1}{8} \cdot 4$
$r = \frac{1}{2}$

Since $| r | < 1$, we may continue...
$a = \frac{1}{4}$

$\frac{a}{1 - r} = \frac{\frac{1}{4}}{1 - \frac{1}{2}}$
$\frac{a}{1 - r} = \frac{\frac{1}{4}}{\frac{1}{2}}$
$\frac{a}{1 - r} = \left(\frac{1}{4}\right) \div \left(\frac{1}{2}\right)$
$\frac{a}{1 - r} = \left(\frac{1}{4}\right) \cdot \left(2\right)$

$\frac{a}{1 - r} = \frac{1}{2}$