Let a 1 = b 1 = 1 , a n = a n - 1 + 2 and b n = a n + b n - 1 for every natural number n ≥ 2 . Then ∑ n = 1 15 a n · b n is equal to _____ .

Step-by-step solution

Given a n - a n - 1 = 2 So the sequence is an A.P. with first term a 1 = 1 and common difference d = 2 i.e., a n = a 1 + n - 1 d = 2 n - 1 Also, a n = b n - b n - 1 We know 2 n - 1 = n 2 - n - 1 2 So, b n = n 2 Now, ∑ n = 1 15 a n · b n = ∑ n = 1 15 2 n 3 - n 2 = 2 ∑ n = 1 15 n 3 - ∑ n = 1 15 n 2 = 2 15 · 16 2 2 - 15 · 16 · 31 6 = 27560