Find the 20^{text{th}} and n^{text{th}} terms | vedclass

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(A) The given $G.P.$ is $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \dots$ Here,the first term $a = \frac{5}{2}$. The common ratio $r = \frac{5/4}{5/2} =

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$\frac{5}{2^{20}}, \frac{5}{2^n}$

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(A) The given $G.P.$ is $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \dots$ Here,the first term $a = \frac{5}{2}$. The common ratio $r = \frac{5/4}{5/2} = \frac{5}{4} 	imes \frac{2}{5} = \frac{1}{2}$. The $n^{	ext{th}}$ term of a $G.P.$ is given by $a_n = a r^{n-1}$. For the $20^{	ext{th}}$ term: $a_{20} = \frac{5}{2} \left(\frac{1}{2}ight)^{20-1} = \frac{5}{2} \left(\frac{1}{2}ight)^{19} = \frac{5}{2^1 	imes 2^{19}} = \frac{5}{2^{20}}$. For the $n^{	ext{th}}$ term: $a_n = \frac{5}{2} \left(\frac{1}{2}ight)^{n-1} = \frac{5}{2^1 	imes 2^{n-1}} = \frac{5}{2^{1+n-1}} = \frac{5}{2^n}$.

Find the $20^{\text{th}}$ and $n^{\text{th}}$ terms of the $G.P.$ $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \dots$

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