Question

Find the sum to n terms of the G.P., whose $k^{th}$ term is $5^k$.

Answer 1

Given:-

$a_k=5^k$

To find:- Sum of $n$ terms $\left(S_n\right)=?$

$a_k=5^k\left(\text{Given}\right)$

Therefore

$a_1=5^1$

$a_2=5^2=25$

$a_3=5^3=125$

The given G.P. will be $5,25,125,.....$.

Therefore,

First term $\left(a\right)=5$

Common ratio $\left(r\right)=\frac{a_2}{a_1}=\frac{25}{5}=5$

As we know that, sum of $n$ terms in a G.P. is given as-

$S_n=\frac{a(r^n-1)}{r-1}$

Therefore, for the given series,

$S_n=\frac{5(5^n-1)}{5-1}$

$\Rightarrow S_n=\frac{5}{4}(5^n-1)$

Hence the sum of $n$ terms of given G.P. is $\frac{5}{4}(5^n-1)$.