Let the coefficient of x r in the expansion of x + 3 n − 1 + x + 3 n − 2 x + 2 + x + 3 n − 3 x + 2 2 + . ... + x + 2 n − 1 be α r . If ∑ r = 0 n α r = β n − γ n , β , γ ∈ N , then the value of β 2 + γ 2 equals _______.

Step-by-step solution

Let, S = x + 3 n − 1 + x + 3 n − 2 x + 2 + x + 3 n − 3 x + 2 2 + . .... + x + 2 n − 1 Now, using the sum of G.P formula we get, ⇒ S = x + 3 n - 1 1 - x + 2 x + 3 n 1 - x + 2 x + 3 ⇒ S = x + 3 n - x + 2 n Now, finding the sum of coefficients we get, ⇒ ∑ α r = ∑ x + 3 n - ∑ x + 2 n ⇒ 4 n − 3 n = β n − γ n by taking x = 1 So, on comparing both side we get, β = 4 , γ = 3 Hence, β 2 + γ 2 = 16 + 9 = 25