Answer
7.7777
Work Step by Step
The series is $7+0.7+0.07+\cdots $.
Here, ${{a}_{1}}=7$, $n=5$. The value of r is calculated as:
$\begin{align}
& r=\frac{0.7}{7} \\
& =0.1
\end{align}$
Substituting ${{a}_{1}}=7$, $n=5$ and $r=0.1$ in ${{S}_{n}}=\frac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{1-r}$, for any $r\ne 1$:
$\begin{align}
& {{S}_{5}}=\frac{7\cdot \left( 1-{{\left( 0.1 \right)}^{5}} \right)}{1-0.1} \\
& =\frac{7\left( 1-0.00001 \right)}{0.9} \\
& =\frac{7\left( 0.99999 \right)}{0.9} \\
& =7.7777
\end{align}$
Thus, the sum of the first nine terms, ${{S}_{5}}$ for $7+0.7+0.07+\cdots $ is 7.7777.
