<s>[INST] Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction I recently proved that
$$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$
using mathematical induction.  I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any other proofs.
 [/INST] $f(n)=1^3+2^3+3^3+\cdots+n^3$
$f(n-1)=1^3+2^3+3^3+\cdots+(n-1)^3$
$f(n)-f(n-1)=n^3$
if $g(n)= (1+2+3+4+\cdots+n)^2$  then
$$g(n)-g(n-1)=(1+2+3+4+\cdots+n)^2-(1+2+3+4+\cdots+(n-1))^2$$
using $a^2-b^2=(a+b)(a-b)$
$$\begin{align}g(n)-g(n-1)&=\\
&=[(1+\dots+n)+(1+\dots +(n-1))][(1+\dots+ n)-(1+\dots+(n-1)]\\
&=[2(1+2+3+4+\cdots+(n-1))+n]n\\
&=\left(2\frac{n(n-1)}{2}+n\right)n\\
&==(n(n-1)+n)n\\
&=n^3
\end{align}$$
So $f(n)$ and $g(n)$ are equal.
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