Let a n n = 1 ∞ be a sequence such that a 1 = 1 , a 2 = 1 and a n + 2 = 2 a n + 1 + a n for all n ≥ 1 . Then the value of 47 ∑ n = 1 ∞ a n 2 3 n is equal to ________.

Step-by-step solution

We have, 47 ∑ n = 1 ∞ a n 2 3 n ⇒ ∑ n = 1 ∞ a n 8 n = P (let) And, a n + 2 = 2 a n + 1 + a n ⇒ a n + 2 8 n = 2 a n + 1 8 n + a n 8 n ⇒ 64 a n + 2 8 n + 2 = 16 a n + 1 8 n + 1 + a n 8 n ⇒ 64 ∑ n = 1 ∞ a n + 2 8 n + 2 = 16 ∑ n = 1 ∞ a n + 1 8 n + 1 + ∑ n = 1 ∞ a n 8 n ⇒ 64 P - a 1 8 - a 2 8 2 = 16 P - a 1 8 + P ⇒ 64 P - 1 8 - 1 64 = 16 P - 1 8 + P ⇒ 64 P - 8 - 1 = 16 P - 2 + P ⇒ 47 P = 7 ⇒ 47 ∑ n =