Let 2 ^ nd , 8 ^ th and 44 ^ th , terms of a non-constant A . P ....

Given: ${2}^{nd}, {8}^{th}, {44}^{th}$ terms of $A . P$ are ${1}^{st}, {2}^{nd} \text{ and } {3}^{rd}$ terms respectively of $G . P .$

And first term of $A . P .$ is $1$ and let $d$ be common difference

So, $1 + d, 1 + 7d, 1 + 43d$ are ${1}^{st}, {2}^{nd} \text{ and } {3}^{rd}$ term of $G . P .$

$⇒ \dfrac{1 + 7d}{1 + d} = \dfrac{1 + 43d}{1 + 7d}$

$⇒ {\left(1 + 7d\right)}^{2} = \left(1 + 43d\right)\left(1 + d\right)$

$⇒ 1 + 49{d}^{2} + 14d = 1 + 43{d}^{2} + 44d$

$⇒ 6{d}^{2} = 30d$

$⇒ d = 5$

Now finding the sum of first $20$ terms of AP,

$⇒ {S}_{20} = 10\left[2 \times 1 + 19 \times 5\right]$

$⇒ {S}_{20} = 970$