If the 2^{nd}, 5^{th}, and 9^{th} terms of a | vedclass

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(D) Let the first term of the $A.P.$ be $a$ and the common difference be $d$. The terms are given by $T_n = a + (n-1)d$.The $2^{nd}, 5^{th},$ and $9^{

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$\frac{4}{3}$

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(D) Let the first term of the $A.P.$ be $a$ and the common difference be $d$. The terms are given by $T_n = a + (n-1)d$.The $2^{nd}, 5^{th},$ and $9^{th}$ terms are $a+d, a+4d,$ and $a+8d$ respectively.Since these terms are in $G.P.$,the square of the middle term equals the product of the first and third terms:$(a+4d)^2 = (a+d)(a+8d)$Expanding both sides:$a^2 + 8ad + 16d^2 = a^2 + 9ad + 8d^2$Subtracting $a^2$ and rearranging terms:$8d^2 = ad$Since the $A.P.$ is non-constant,$d 
eq 0$,so we can divide by $d$:$a = 8d$The terms of the $G.P.$ are:$T_2 = a+d = 8d+d = 9d$$T_5 = a+4d = 8d+4d = 12d$$T_9 = a+8d = 8d+8d = 16d$The common ratio $r$ is given by:$r = \frac{T_5}{T_2} = \frac{12d}{9d} = \frac{4}{3}$

If the $2^{nd}, 5^{th},$ and $9^{th}$ terms of a non-constant $A.P.$ are in $G.P.$,then the common ratio of this $G.P.$ is:

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