Answer
$\frac{4}{33}$
Work Step by Step
$0.12121212\ldots $
This can be written as,
$0.12121212\ldots =0.12+0.0012+0.000012+\cdots $
This is an infinite geometric series:
${{a}_{1}}=0.12$ and ${{a}_{2}}=0.0012$.
So, the value of $\left| r \right|$ is,
$\begin{align}
& \left| r \right|=\left| \frac{0.0012}{0.12} \right| \\
& =\left| 0.01 \right| \\
& =0.01
\end{align}$
We know that
${{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}$.
$\begin{align}
& {{S}_{\infty }}=\frac{{{a}_{1}}}{1-r} \\
& =\frac{0.12}{1-0.01} \\
& =\frac{0.12}{0.99} \\
& =\frac{4}{33}
\end{align}$
Thus, the fraction notation of the decimal number $0.12121212\ldots $ is $\frac{4}{33}$.
