Question

What is the sum of the first five terms of a geometric series with a1= 10 and r= 1/5

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}\\----------\\a_1=10\\r=(1)/(5)\\n=5\end{cases}`

`\bf S_5=10\left( \cfrac{1-\left( (1)/(5) \right)^5}{1-\left( (1)/(5) \right)} \right)\implies S_5=10\left( \cfrac{1-(1^5)/(5^5)}{(4)/(5)}\right)\implies S_5=10\left( \cfrac{(3124)/(3125)}{(4)/(5)} \right)\\\\\\S_5=10\cdot \cfrac{781}{625}\implies S_5=\cfrac{1562}{125}`