2^ 3 n -7 n-1 is divisible by

Step-by-step solution

Let f(n) = 2^ 3n - 7n - 1 = 8^n - 7n - 1 . For n = 1 , f(1) = 8^1 - 7(1) - 1 = 8 - 8 = 0 , which is divisible by any integer. For n = 2 , f(2) = 8^2 - 7(2) - 1 = 64 - 14 - 1 = 49 . For n = 3 , f(3) = 8^3 - 7(3) - 1 = 512 - 21 - 1 = 490 . Since f(2) = 49 and f(3) = 490 = 10 49 , the expression 2^ 3n - 7n - 1 is divisible by 49 for all n 1 . Using the binomial expansion, 8^n = (1 + 7)^n = 1 + ^ n C₁(7) + ^ n C₂(7^2) + ^ n C₃(7^3) + + ^ n C_ n (7^n) . Substituting this into the expression: f(n) = (1 + 7n + ^ n C₂(49)