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.

Solve (x - 1)^n =x^n , where n is a positive integer.

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

The sum of the coefficients of (5x-4y)^n where n is positive integer, is

Prove that 2^(n) gt n for all positive integers n.

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.

If a, b, c, d be 3^(rd), 4^(th), 5^(th) and 6^(th) terms of the expansion of (p+q)^(n) , where n is a positive integer prove that (b^(2)-ac)/(c^(2)-b)=(5a)/(3c) .

If NN denotes the set of all positive integers and if f:NN rarrN is defined by f(n)= the sum of positive divisors of n then f(2^(k).3) , where k is a positive integer is

Using binomial theorem, prove that 5^(4n)+52n-1 is divisible by 676 for all positive integers n.