If a and b are distinct integers, prove that a-b is a factor of `a^n` - `b^n` , whenever n is a positive integer.

Prove that the function f(x)= x^(n) is continuous at x= n, where n is a positive integer.

Prove that int_(0)^(infty)x^n e^(-x) dx=n!., Where n is a positive integer.

Prove that 2^n >1+nsqrt(2^(n-1)),AAn >2 where n is a positive integer.

Write the converse of the following statements. (i) If a number n is even, then n^(2) is even. (ii) If you do all the exercises in the book, you get an A grade in the class. (iii) If two integers a and b are such that a gt b , then a -b is always a positive integer.

The product of n positive numbers is 1, then their sum is a positive integer, that is

Evaluate int_(0)^(infty)(x^(n))/(n^(x)) dx, where n is a positive integer.

Derivative of f(x)=x^n is nx^(n-1) for any positive integer n.