If 4^{th} and 8^{th} terms of a G.P. are 24 a | vedclass

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(C) 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 $4^{

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$3, 2$

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(C) 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 $4^{th}$ term is $24$,we have:$a r^{4-1} = 24 \Rightarrow a r^3 = 24$ $...(1)$Given that the $8^{th}$ term is $384$,we have:$a r^{8-1} = 384 \Rightarrow a r^7 = 384$ $...(2)$Dividing equation $(2)$ by equation $(1)$:$\frac{a r^7}{a r^3} = \frac{384}{24}$$r^4 = 16$$r^4 = 2^4 \Rightarrow r = 2$ (Taking the positive real root for common ratio).Substituting $r = 2$ into equation $(1)$:$a(2)^3 = 24$$a(8) = 24$$a = \frac{24}{8} = 3$Thus,the first term is $3$ and the common ratio is $2$.

If $4^{th}$ and $8^{th}$ terms of a $G.P.$ are $24$ and $384$ respectively,then find out the first term and common ratio.

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