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

The Correct Option is D

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