Question

The sum of first $n$ terms of a sequence is $\frac{6^n-5^n}{5^n}$
Find its $n^{th}$ term and examine whether the sequence is an A.P of G.P.

Answer 1

Sum of $n$ terms $=S_n=\frac{6^n-5^n}{5^n}={\left(\frac{6}{5}\right)}^n-1$

Sum of $(n-1)$ terms $=S_{n-1}={\left(\frac{6}{5}\right)}^{n-1}-1$

So, $n^{th}$ term $=S_n-S_{n-1}={\left(\frac{6}{5}\right)}^n-{\left(\frac{6}{5}\right)}^{n-1}=\frac{1}{6}{\left(\frac{6}{5}\right)}^n$

So, this sequence is a G.P with common ratio $=\frac{6}{5}$