Answer
$\frac{7}{9}$
Work Step by Step
We express the number as a sum of fractions:
$0.777... = \frac{7}{10}+\frac{7}{100}+\frac{7}{1000}$...
We know that this represents an infinite geometric series with $a= \frac{7}{10}$ and $r= \frac{1}{10}$.
We know the sum of an infinite geometric series is:
$S_{\infty}=\frac{a}{1-r}$
$S_{\infty}=\frac{\frac{7}{10}}{1-\frac{1}{10}}=\frac{7}{9}$
