Answer
$\frac{5}{33}$
Work Step by Step
$0.15151515\ldots $
This can be written as,
$0.151515151\ldots =0.15+0.0015+0.000015+\cdots $
This is an infinite geometric series:
${{a}_{1}}=0.15$ and ${{a}_{2}}=0.0015$.
So, the value of $\left| r \right|$ is,
$\begin{align}
& \left| r \right|=\left| \frac{0.0015}{0.15} \right| \\
& =\left| 0.01 \right| \\
& =0.01
\end{align}$
${{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}$.
$\begin{align}
& {{S}_{\infty }}=\frac{{{a}_{1}}}{1-r} \\
& =\frac{0.15}{1-0.01} \\
& =\frac{0.15}{0.99} \\
& =\frac{5}{33}
\end{align}$
Thus, the fraction notation of the decimal number $0.15151515\ldots $ is $\frac{5}{33}$.
