Find the $20^{\text {th }}$ and $n^{\text {th }}$ terms of the $G.P.$ $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots$
Find the $20^{\text {th }}$ and $n^{\text {th }}$ terms of the $G.P.$ $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots$
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 $=\frac{5 / 4}{5 / 2}=\frac{1}{2}$
$a_{20}=a r^{20-1}=\frac{5}{2}\left(\frac{1}{2}\right)^{19}=\frac{5}{(2)(2)^{19}}=\frac{5}{(2)^{20}}$
$a_{n}=a r^{n-1}=\frac{5}{2}\left(\frac{1}{2}\right)^{n-1}=\frac{5}{(2)(2)^{n-1}}=\frac{5}{(2)^{n}}$
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