Question:
Find the $20^{\text {th }}$ and $n^{\text {thterms }}$ of the G.P. $=\frac{5}{2}, \frac{5}{4}, \frac{5}{8} \ldots$
Solution:
The given G.P. is $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots$
Here, $a=$ First term $=\frac{5}{2}$
$r=$ Common ratio $=\frac{\frac{5}{4}}{\frac{5}{2}}=\frac{1}{2}$
$a_{20}=a r^{20-1}=\frac{5}{2}\left(\frac{1}{2}\right)^{19}=\frac{5}{(2)(2)^{19}}=\frac{5}{(2)^{20}}$
$a_{n}=a r^{n-1}=\frac{5}{2}\left(\frac{1}{2}\right)^{n-1}=\frac{5}{(2)(2)^{n-1}}=\frac{5}{(2)^{n}}$