Question

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

• A. 980
• B. 960
• C. 990
• D. 970

Answer 1

Answer: (D)

970

Answer 2

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

$\text{2nd term} = 1 + d, \quad \text{8th term} = 1 + 7d, \quad \text{44th term} = 1 + 43d$

These terms are in G.P., so:

$(1 + 7d)^2 = (1 + d)(1 + 43d)$

Expanding and simplifying:

$1 + 49d^2 + 14d = 1 + 44d + 43d^2$ $6d^2 - 30d = 0$ $d = 5$

The sum of the first 20 terms of the A.P. is:

$S_{20} = \frac{20}{2} \left[ 2 \cdot 1 + (20 - 1) \cdot 5 \right]$ $= 10 \cdot (2 + 95) = 10 \cdot 97 = 970$

Thus, the answer is:

970