Answer
See below.
Work Step by Step
Proofs using mathematical induction consist of two steps:
1) The base case: here we prove that the statement holds for the first natural number.
2) The inductive step: assume the statement is true for some arbitrary natural number $n$ larger than the first natural number; then we prove that then the statement also holds for $n + 1$.
Hence, here we have:
1) For $n=1: 1^3=\frac{1^2(1+1)^2)}{4}$.
2) Assume for $n=k: 1^3+2^3+k^3=\frac{k^2(k+1)^2}{4}$. Then for $n=k+1$:
$1^3+2^3+k^3+(k+1)^3=\frac{k^2(k+1)^2}{4}+(k+1)^3=\frac{k^2(k+1)^2}{4}+k^3+3k^2+3k+1=\frac{(k+1)^2(k+2)^2)}{4}=\frac{(k+1)^2(k+1+1)^2}{4}.$
Thus we proved what we wanted to.