[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\\r=\frac{1}{5}\\n=5\end{cases} [/tex]
[tex]\bf S_5=10\left( \cfrac{1-\left( \frac{1}{5} \right)^5}{1-\left( \frac{1}{5} \right)} \right)\implies S_5=10\left( \cfrac{1-\frac{1^5}{5^5}}{\frac{4}{5}}\right)\implies S_5=10\left( \cfrac{\frac{3124}{3125}}{\frac{4}{5}} \right)\\\\\\S_5=10\cdot \cfrac{781}{625}\implies S_5=\cfrac{1562}{125} [/tex]
