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

$\Rightarrow \frac{1+7d}{1+d}=\frac{{\displaystyle 1+43d}}{{\displaystyle 1+7d}}$

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

$\Rightarrow 1+49d^2+14d=1+43d^2+44d$

$\Rightarrow 6d^2=30d$

$\Rightarrow d=5$

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

$\Rightarrow S_{20}=10\left[2\times 1+19\times 5\right]$

$\Rightarrow S_{20}=970$