1. Write the A.P. terms

Let the A.P. have first term $a=1$ and common difference $d$.

So its $n$th term is $T_n = a + (n-1)d = 1+(n-1)d.$

Hence,

• $2$nd term $= 1+d$
• $8$th term $= 1+7d$
• $44$th term $= 1+43d$

These are given to be the first three terms of a G.P.

2. Use the G.P. condition

If three numbers are in G.P., then the square of the middle term equals the product of the other two: $(1+7d)^2 = (1+d)(1+43d).$

Expand both sides: $1+14d+49d^2 = 1+44d+43d^2.$

Now simplify: $14d+49d^2 = 44d+43d^2$ $6d^2-30d=0$ $6d(d-5)=0.$

So, $d=0 \quad \text{or} \quad d=5.$

Since the A.P. is non-constant, $d \ne 0$. Therefore, $d=5.$

3. Find the sum of first 20 terms

Sum of first $n$ terms of an A.P. is $S_n = \frac{n}{2}\left[2a+(n-1)d\right].$

Here, $a=1$, $d=5$, $n=20$: $S_{20} = \frac{20}{2}\left[2(1)+19(5)\right]$ $=10(2+95)$ $=10\cdot 97$ $=970.$

4. Check options

$970$ corresponds to Option D.

Final Answer: $\boxed{970}$