Question: Is there an intuitive way to get cubic sum? From this post: [combination of quadratic and cubic series](https://math.stackexchange.com/questions/876873/combination-of-quadratic-and-cubic-series/876875#876875) and Wikipedia: [Faulhaber formula](http://en.wikipedia.org/wiki/Faulhaber%27s_formula), I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$
I think the cubic sum is squaring the arithmetic sum $$1^3 + 2^3 + \dots + n^3 = (1 + 2 + \dots + n)^2$$
But how to prove it? Please help me. Grazie!

Answer: 
Maybe this will help you visualize it:

![](https://i.stack.imgur.com/MXOt5.png)

[Source.](http://en.wikipedia.org/wiki/File:Sum_of_cubes.png)

or this one which is clearer:

$\phantom{XXXXXXXX}$![e](https://i.stack.imgur.com/MjQGP.jpg)
We have
$$
\sum\_{k=1}^{n}k^3 = 1 + 8 + 27 + \ldots + n^3 = \\
\underbrace{1}\_{1^3} + \underbrace{3+5}\_{2^3} + \underbrace{7 + 9 + 11}\_{3^3} + \underbrace{13 + 15 + 17 + 19}\_{4^3} + \ldots = \\
\underbrace{\underbrace{\underbrace{1}\_{1^2} + 3}\_{2^2} + 5}\_{3^2} + \ldots
$$
which is
$$
\left( \sum\_{k=1}^{n}k \right)^2
$$