The sum of some terms of $G.P.$ is $315$ whose first term and the common ratio are $5$ and $2,$ respectively. Find the last term and the number of terms.
The sum of some terms of $G.P.$ is $315$ whose first term and the common ratio are $5$ and $2,$ respectively. Find the last term and the number of terms.
Let the sum of n terms of the $G.P.$ be $315$
It is known that, $S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}$
It is given that the first term $a$ is $5$ and common ratio $r$ is $2$
$\therefore 315=\frac{5\left(2^{n}-1\right)}{2-1}$
$\Rightarrow 2^{n}-1=63$
$\Rightarrow 2^{n}=64=(2)^{6}$
$\Rightarrow n=6$
$\therefore$ Last term of the $G.P.$ $=6^{\text {th }}$ term $=a r^{6-1}=(5)(2)^{5}=(5)(32)$
$=160$
Thus, the last term of the $G.P.$ is $160 .$
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