Question

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 .

Answer 1

Let the sum of $n$ terms of the G.P. be $315.$
It is known that, $S_n=\frac{a(r^n-1)}{r-1}$
It is given that the first term $a$ is 5 and common ratio $r$ is 2.
$\therefore 315=\frac{5(2^n-1)}{2-1}\Rightarrow 2^n-1=63\Rightarrow 2^n=64=2^6\Rightarrow n=6$
$\therefore$ Last term of the G.P.:

$6^{th}$ term $={ar}^{6-1}=\left(5\right){\left(2\right)}^5=\left(5\right)\left(32\right)=160.$
Thus, the last term of the G.P. is $160.$