Question

What is the sum of the first 5 terms of 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\\ n=5\\ r=(1)/(5) \end{cases}`


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