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

Prove, by Mathematical Induction, that for all n in N, 2.7^n+3.5^n-5 is divisible by 24 .

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

Prove, by Mathematical Induction, that for all n in N, n(n + 1) (n + 2) (n + 3) is a multiple of 24.

Prove by Induction, for all n in N : 2^n >n .

Prove by Induction, for all n in N : 2^n < 3^n .

Prove by Induction, for all n in N : (n+3)^2 le 2^(n+3) .

Prove by Induction, that (2n+7)le (n+3)^2 for all n in N. Using this, prove by induction that : (n+3)^2 le 2^(n+3) for all n in N.

State the converses of the following statement. In each case, also decide whether the converse is true or false.:- If n is an even integer, then 2n + 1 is an odd integer.

By the Principle of Mathematical Induction, prove that for all n in N, 3^(2n) when divided by 8, the remainder is 1 always.