<s>[INST] Intuitive ways to get formula of cubic sum Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber 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!
 [/INST] 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
$$
\left( \sum_{k=1}^{n}k \right)^2
$$
</s>