## College Algebra 7th Edition

$\frac{7}{9}$
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: $S_{\infty}=\frac{a}{1-r}$ $S_{\infty}=\frac{\frac{7}{10}}{1-\frac{1}{10}}=\frac{7}{9}$