# 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!

## 2 Answers

Maybe this will help you visualize it: Source.

or this one which is clearer:

$$\phantom{XXXXXXXX}$$ • Very nice! Thanks for your answer :) – L Lawliet Jul 24 '14 at 14:01
• @LLawliet You're welcome, glad I could help! ;) – Hakim Jul 24 '14 at 14:01

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 $$\left( \sum_{k=1}^{n}k \right)^2$$

• This is also very nice! Thanks! – L Lawliet Jul 24 '14 at 14:05
• Yes, I find it quite amusing, just messing around with numbers and see what comes out =) – Noxet Jul 25 '14 at 8:34