# How do you find the sum of the infinite geometric series 5 + 5/3 + 5/9 + 5/27 +....?

The sum is $\frac{15}{2}$

#### Explanation:

You can rewrite this as follows

$5 + \frac{5}{3} + \frac{5}{9} + \frac{5}{27} + \ldots = 5 \cdot \left(1 + \frac{1}{3} + \frac{1}{3} ^ 2 + \frac{1}{3} ^ 3 + \ldots\right)$

Now the sum $1 + \frac{1}{3} + \frac{1}{3} ^ 2 + \frac{1}{3} ^ 3 + \ldots$ is given by the formula

$S = \frac{{a}_{1}}{1 - r}$

where ${a}_{1}$ is the first term of the series and $r$ is the ratio of successive terms in our case $r = \frac{1}{3}$

Hence

$S = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2}$