The 5^{th} term of a G.P. is 2, find the prod | vedclass

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(B) Let $a$ be the first term and $r$ be the common ratio of the $G.P.$The $n^{th}$ term of a $G.P.$ is given by $a_n = a r^{n-1}$.Given that the $5^{

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$512$

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(B) Let $a$ be the first term and $r$ be the common ratio of the $G.P.$The $n^{th}$ term of a $G.P.$ is given by $a_n = a r^{n-1}$.Given that the $5^{th}$ term is $2$, we have:$a_5 = a r^{5-1} = a r^4 = 2$ $...(1)$We need to find the product of the first $9$ terms:$P = a 	imes (a r) 	imes (a r^2) 	imes \dots 	imes (a r^8)$$P = a^9 	imes r^{(1 + 2 + 3 + \dots + 8)}$The sum of the first $n$ natural numbers is given by $\frac{n(n+1)}{2}$.For $n=8$, the sum is $\frac{8 	imes 9}{2} = 36$.Therefore, $P = a^9 	imes r^{36} = (a r^4)^9$.Substituting the value from equation $(1)$:$P = (2)^9 = 512$.

The $5^{th}$ term of a $G.P.$ is $2,$ find the product of first $9$ terms.

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