# How do you find the sum of the following infinite geometric series, if it exists. 2 + 1.5 + 1.125 + 0.8437 +…?

Dec 19, 2015

Sum of the given infinite geometric series is $8$

#### Explanation:

The ratio, $r$ between successive geometric terms is
$\textcolor{w h i t e}{\text{XXX}} \frac{1.5}{2} = \frac{1.125}{1.5} = \ldots = 0.75$

The initial term ${a}_{1}$ is given as $2$

Since $\left\mid r \right\mid < 1$
this infinite geometric series has a sum given by the formula
$\textcolor{w h i t e}{\text{XXX}} \frac{{a}_{1}}{1 - r}$

In this case
sum $= \frac{2}{\left(1 - 0.75\right)} = \frac{2}{0.25} = 8$