Let 2^{text{nd}},8^{text{th}},and 44^{text{th | vedclass

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(D) Let the first term of the $A.P.$ be $a = 1$ and the common difference be $d$.The $2^{	ext{nd}}$,$8^{	ext{th}}$,and $44^{	ext{th}}$ terms are $1

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$970$

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(D) Let the first term of the $A.P.$ be $a = 1$ and the common difference be $d$.The $2^{	ext{nd}}$,$8^{	ext{th}}$,and $44^{	ext{th}}$ terms are $1+d$,$1+7d$,and $1+43d$ respectively.Since these terms are in $G.P.$,we have $(1+7d)^2 = (1+d)(1+43d)$.Expanding both sides: $1 + 49d^2 + 14d = 1 + 44d + 43d^2$.Simplifying: $6d^2 - 30d = 0$,which gives $6d(d - 5) = 0$.Since the $A.P.$ is non-constant,$d 
eq 0$,so $d = 5$.The sum of the first $20$ terms $S_{20} = \frac{20}{2}[2(1) + (20-1)5]$.$S_{20} = 10[2 + 95] = 10 	imes 97 = 970$.

Let $2^{\text{nd}}$,$8^{\text{th}}$,and $44^{\text{th}}$ terms of a non-constant $A.P.$ be respectively the $1^{\text{st}}$,$2^{\text{nd}}$,and $3^{\text{rd}}$ terms of a $G.P.$ If the first term of the $A.P.$ is $1$,then the sum of the first $20$ terms is equal to:

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