Prove the following by using the principle of mathematical induction for all

Let us denote the given equality by P(n), i.e.,

\( 1^3+2^3+3^3+.......+n^3=(\dfrac{n(n+1)}{2})^2\)

For n=1,
L.H.S. 1³=1

R.H.S.= \(( \dfrac{1(1+1)}{2})^2\)

= (1)²= 1

Therefore,P(n) is true for n=1.

Let us assume that P (k) is true for some positive integer k , i.e.,

\( 1^3+2^3+3^3+.......+k^3=(\dfrac{k(k+1)}{2})^2\) ..........(i)

Now, we have to prove that P(k+1) is also true.

Consider

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

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

\( (\dfrac{k(k+1)}{2})^2+(k+1)^3\)......... [Using (i)]

\( (\dfrac{k^2(k+1)^2}{4})^2+(k+1)^3\)

\( \dfrac{k^2(k+1)^2+4(k+1)^3}{4}\)

= \( \dfrac{(k+1)^2(k^2+4(k+1)}{4}\)

= \( \dfrac{(k+1)^2(k+2)^2}{4}\)

= \( \dfrac{(k+1)^2(k+1+1)^2}{4}\)

= \( \dfrac{(k+1)(k+1+1)^2}{2}\)

Therefore,P (k+1)  holds whenever  P(k) holds.

Hence, the given equality is true for all natural numbers i.e., N by the principle of mathematical induction.