Prove the following by the method of induction for all `n in N`: `1^3 + 3^3 + 5^3 +` ... to n terms `= n^2 (2n^2 - 1)`.

Prove the following by the method of induction for all n in N : 1.3 + 3.5 + 5.7 +... to n terms =n/3 (4n^2 + 6n - 1) .

Prove the following by the method of induction for all n in N : 3 + 7+ 11 +.. to n terms = n(2n + 1).

Prove the following by the method of induction for all n in N : 2 + 4 + 6 +... +2n = n(n + 1).

Prove the following by the method of induction for all n in N : 1^2 + 3^2 + 5^2 + ... + (2n - 1)^2 = n/3 (2n -1) (2n +1)

Prove the following by the method of induction for all n in N : 5 + 5^2 + 5^3 + .. + 5^n = 5/4 (5^n - 1)

Prove the following by the method of induction for all n in N : 3^n - 2n - 1 s divisible by 4.

Prove the following by the method of induction for all n in N : 1^2 + 2^2 + 3^2 + ... + n^2 = (n(n + 1) (2n + 1)) / 6 .

Prove the following by the method of induction for all n in N : 1.2 + 2.3 + 3.4 +...+ n(n + 1) = n/3 (n+1)(n+2).

Prove the following by the method of induction for all n in N : (2^(3n) -1) is divisible by 7.

Prove the following by the method of induction for all n in N : (2^(4n) -1) is divisible by 15.