`(-2)^(31)` is a positive integer.

Write the converse of the following statement : If two integers a and b are such that a > b, then (a-b) is always a positive integer.

If a_1=1 and a_n =na_(n-1) , for all positive integers n ge 2 , then a_5 is equal to :

Using mathematical induction prove that d(x^n)/dx = nx^(n-1) for all positive integers n.

Write the middle term in the expansion of : (1+x)^(2n) , where n is a positive integer.

Prove that (n!)^2 le n^n. (n!)<(2n)! for all positive integers n.

Let m be the smallest positive integer such that the coefficient of x^(2) in the expansion of (1+x)^(2)+(1+x)^(3) + "……." + (1+x)^(49) + (1+mx)^(50) is (3n+1) .^(51)C_(3) for some positive integer n, then the value of n is "_____" .

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. [Hint write a^n = (a - b + b)^n and expand]

Using Permutation or otherwise, prove that ((n^2)!)/((n!)^2) is an integer, where n is a positive integer.

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.