Envisioning (1³ + ... + n³) = (1 + ... + n)²

Question

Is there a way to geomtrically visualize the identity (1³ + 2³ + ... + n³) = (1 + 2 + ... + n)², n ∈ ℕ? Are there any other similar identities?

Have you proven the identity you reference, or seen its proof? (It, indeed, holds, e.g. see this. — amWhy, Nov 09 '16 at 00:12
As for a geometric visualization see AOPS. Once there, scroll down until you see Nichomauss' Theorem. Also see the applet here — amWhy, Nov 09 '16 at 00:15

score 1 · Accepted Answer · edited Apr 13 '17 at 12:20

1

There is a nice explanation and diagram on Wikipedia.

Another similar identity is found in the question here

$$\sigma_4=\sigma_2\frac{6\sigma_1-1}5$$ where $\sigma_m=\sum_{r=1}^n r^m$.

There are more. These can be found in the paper here.

But none as neat as the sum of cubes being the square of the sum of integers. :)

edited Apr 13 '17 at 12:20

Community

answered Nov 09 '16 at 02:40

Hypergeometricx

1 Answers1