Using Induction prove the given statement;

Question

By using the Principle Of Mathematical Induction prove that:$1^3+2^3+3^3+.......+n^3=[\frac {n(n+1)}{2}]^2$.

My Approach:

Let, $P(n): 1^3+2^3+3^3+.....+k^3=[\frac {n(n+1)}{2}]^2$.

Base case $(n=1)$ $$L.H.S=1$$ $$R.H.S=[\frac {1(1+1)}{2}]^2$$ $$=[\frac {1\times 2}{2}]^2$$ $$=1$$.

$i.e., L.H.S=R.H.S$. So, $P(1)$ is true.

Induction Hypothesis:$(let, n=k)$.

Assume $P(k): 1^3+2^3+3^3+....+k^3=[\frac {k(k+1)}{2}]^2$ is true.

Please help to continue from here.

Answer 1

Add $(n+1)^3$ to both sides to get $ 1^3+2^3+...+n^3+(n+1)^3 = \frac{([n(n+1)]^2}{4} + (n+1)^3 = \frac{(n+1)(n^2(n+1)+4(n+1)^2)}{4} = \frac{[(n+1)(n+2)]^2}{4} $