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 the A. P. is 1, then the sum of its first 20 terms is equal to -

Solution

Maths sir · 4 steps

Write the three AP terms with d, then use the GP rule: middle squared equals product of the ends.

1. Terms in d. The first term is 1. Call the common difference d. Then the AP terms are a₂ = 1 + d, a₈ = 1 + 7d and a₄₄ = 1 + 43d.

   (1 + 7d)² = (1 + d)(1 + 43d)

2. Solve for d. Expanding gives 1 + 14d + 49d² = 1 + 44d + 43d², so 6d² = 30d, that is 6d(d - 5) = 0. The AP is non-constant, so d ≠ 0. That leaves d = 5.
3. Sum of 20 terms. S₂₀ = 20/2[2(1) + 19(5)] = 10 × 97 = 970.
4. So. The sum of the first 20 terms is 970, option (d).

The idea to keep. When AP terms form a GP, write each term with a and d and use b² = ac. The word non-constant is there to rule out the root d = 0.

Answer from the NTA final answer key. Solution by PrepWiser.