Mathematcal Induction

For P(1):

Put n=1

RHS =LHS

Hence our assumption is correct for n=1

Assumption P(K):

Let the above statement be correct for n=k

Hence

1^3 + 2^3 + ... + k^3 = [k(k+1)/2]^2.............................1

Prove it For P(k+1):

So LHS will be

1^3 + 2^3 + ... + k^3 + (k+1)^3

Now for terms till n=k put the values obtained in (1)

[k(k+1)/2]^2 + (k+1)^3

On simplifying the above thing you will get

[(k+1)(k+2)/2]^2

This proves that the statement is correct for n = (k+1) when its correct for n= k

Hence through the principle of mathematical induction we have proved that the statement is correct