Question

Prove each of the following statements is true for all positive integers using mathematical induction. Please utilize the structure, steps, and terminology demonstrated in class.

Answer 1

Here we have to prove n! < n^n, for n greaterthanorequalto 2 For n = 2, 2! = 2.1 = 2 2^2 = 4 therefore 2! < 2^2 therefore The statement n! < n^n is true for n = 2 Assume that the statement is true for n = k i.e k! < k^k for k > 2 we have to prove the statement is true for n = k + 1 (k + 1)! = (k + 1)k! < (k + 1) k^k < (k + 1) (k + 1)^k = (k + 1)^k + 1 ie (k + 1)! < (k + 1)^k + 1 therefore statement is true for n = k + 1 therefore By mathematical Induction, the statement n! < n^n for n greaterthanorequalto 2 is true for all positive integral values of n.