Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series 1 2 + 2 · 2 2 + 3 2 + 2 · 4 2 + 5 2 + 2 · 6 2 + … If B - 2 A = 100 λ , then λ is equal to :

Step-by-step solution

Given series, 1 2 + 2 · 2 2 + 3 2 + 2 · 4 2 + 5 2 + 2 · 6 2 + …   Here, A = 1 2 + 2 · 2 2 + 3 2 + 2 · 4 2 + . . . 20 terms = 1 2 +   2 2 + 3 2 + 4 2 + . . . 20   terms + 2 2 + 4 2 +   6 2 + . . . 10   terms = 20 · 21 · 41 6   + 4 10 · 11 · 21 6 ⇒ A = 4410 Similarly for B = 1 2 + 2 2 + 3 3 + . . . 40   terms + 2 2 + 4 2 + 6 2 + . . . 20   terms ⇒ B = 40 × 41 × 81 6 + 4 × 20 × 21 × 41 6 &#86