Prove by direct method that for any integer `n,n^(3)-n` is always even.

Using contrapositive method prove that, if n^(2) is an even integer , then n is also an even integer.

Using contrapositive method prove that , if n^(2) is an even integer, then n is also an even integer.

Prove that 2^n > n for all positive integers n.

Prove by induction that if n is a positive integer not divisible by 3 , then 3^(2n)+3^(n)+1 is divisible by 13 .

Let A ,\ B be two matrices such that they commute. Show that for any positive integer n , A B^n=B^n A

Let A ,B be two matrices such that they commute. Show that for any positive integer n , A B^n=B^n A

Prove by the principle of mathematical induction that for all n in N ,n^2+n is even natural number.

Prove by the principle of mathematical induction that for all n in N ,n^2+n is even natural number.

Let A ,\ B be two matrices such that they commute. Show that for any positive integer n , (A B)^n=A^n B^n

Let A ,B be two matrices such that they commute. Show that for any positive integer n , (i) A B^n=B^n A (ii) (A B)^n=A^nB^n