[tex]\bf \qquad \qquad \textit{sum of a finite geometric sequence}
\\\\
S_n=\sum\limits_{i=1}^{n}\ a_1\cdot r^{i-1}\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad
\begin{cases}
n=n^{th}\ term\\
a_1=\textit{first term's value}\\
r=\textit{common ratio}\\
----------\\
a_1=10\\
n=5\\
r=\frac{1}{5}
\end{cases}[/tex]
[tex]\bf S_5=10\left( \cfrac{1-\left( \frac{1}{5} \right)^5}{1-\frac{1}{5}} \right)\implies S_5=10\left( \cfrac{1-\frac{1}{3125}}{\frac{4}{5}} \right)
\\\\\\
S_5=10\left( \cfrac{\frac{3124}{3125}}{\frac{4}{5}} \right)\implies S_5=10\cdot \cfrac{781}{625}\implies S_5=\cfrac{1562}{125}[/tex]
