The sum of the first 20 terms of the series 5+11+19+29+41+… is :

Solution

Maths sir · 4 steps

The differences 6, 8, 10, 12 form an AP, so the general term is a quadratic in n. Find it first.

1. Look at differences. The gaps between terms are 6, 8, 10, 12, …, an AP with common difference 2. So Tₙ equals 5 plus the sum of the first n-1 gaps.
2. General term. The first n-1 gaps add to (n-1)/2[12 + 2(n-2)] = (n-1)(n+4) = n² + 3n - 4. So Tₙ = n² + 3n + 1. Check: T₁ = 5, T₂ = 11, T₃ = 19.

   S₂₀ = Σₙ₌₁²⁰ n² + 3Σₙ₌₁²⁰ n + 20 = (20 × 21 × 41)/6 + 3 × (20 × 21)/2 + 20

3. Add the parts. Σ n² = 2870, 3Σ n = 3 × 210 = 630, and the constant 1 added 20 times gives 20. Total: 2870 + 630 + 20 = 3520.
4. So. The sum of the first 20 terms is 3520, option (d).

The idea to keep. When the differences of a series form an AP, the general term is a quadratic an² + bn + c. Find Tₙ, then sum it using Σ n², Σ n and Σ 1.

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