Answer
$\frac{2}{11}$
Work Step by Step
The repeating decimal 0.18181818 . . . can be expressed as the infinite geometric series:
$0.1818181818...=0.18+0.18(\frac{1}{100})+0.18(\frac{1}{100})^{2}+0.18(\frac{1}{100})^{3}...$
$a=0.18$
$r=\frac{1}{100}$
$\Sigma=\frac{0.18}{1-\frac{1}{100}}=\frac{2}{11}$
and $\frac{2}{11}$ is also $0.1818181818...$
