Question

How many terms are there in a geometric series if the first term is 5, the common ratio is 2, and the sum of the series is 315?

Answer 1

$\bf \qquad \qquad \textit{sum of a finite geometric sequence} \\\\ S_n=\sum\limits_{i=1}^{n}\ a_1\cdot r^{i-1}\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ a_1=5\\ r=2\\ S_n=315 \end{cases} \\\\\\ 315=5\left( \cfrac{1-2^n}{1-2} \right)\implies \cfrac{315}{5}=\cfrac{1-2^n}{-1}\implies 63=\cfrac{1-2^n}{-1} \\\\\\ -63=1-2^n\implies 2^n=64\qquad \boxed{64=2^6}\qquad 2^n=2^6\implies n=6$