Question

Find the sum of the geometric series:

Answer 1

$\bf \qquad \qquad \textit{sum of a finite geometric sequence} \\\\ S_n=\sum\limits_{i=1}^{n}\ a_1\cdot r^{i-1}\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio} \end{cases}\\\\ -------------------------------$

$\bf \sum\limits_{i=1}^{5}~\stackrel{a_1}{375}\stackrel{r}{\left( \frac{-1}{5} \right)}^{i-1}\qquad \begin{cases} a_1=375\\ r=-\frac{1}{5} \end{cases}\implies S_5=375\left( \cfrac{1-\left( \frac{-1}{5}\right)^5}{1-\left( \frac{-1}{5}\right)} \right)$

$\bf S_5=375\left( \cfrac{1-\left( \frac{-1}{3125} \right)}{1+\frac{1}{5}} \right)\implies S_5=375\left( \cfrac{1+\frac{1}{3125}}{1+\frac{1}{5}} \right) \\\\\\ S_5=375\left( \cfrac{\frac{3126}{3125}}{\frac{6}{5}} \right)\implies S_5=375\left(\cfrac{3126}{3125}\cdot \cfrac{5}{6}\right)\implies S_5=375\left(\cfrac{521}{625}\right) \\\\\\ S_5=\cfrac{1563}{5}\implies S_5=312\frac{3}{5}$