Answer
$130.264\text{ inches}$.
Work Step by Step
From the given information, we can observe that it is a geometric series:
$20,0.9\left( 20 \right),{{0.9}^{2}}\left( 20 \right),{{0.9}^{3}}\left( 20 \right)\cdots $
Here ${{a}_{1}}=20$ and common ratio $r=0\cdot 9$ and $n=10$.
We use the formula for the sum of a geometric series with finite terms ${{S}_{n}}=\frac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{\left( 1-r \right)}$.
Thus
$\begin{align}
& {{S}_{n}}=\frac{20\left( 1-0\cdot {{9}^{10}} \right)}{\left( 1-0\cdot 9 \right)} \\
& \frac{20\left( 1-0\cdot 3486 \right)}{\left( 0\cdot 1 \right)} \\
& =130.264
\end{align}$
