Sum of Sequence of Cubes/Visual Demonstration

Proof
A visual illustration of the proof for $n = 5$:


 * SumOfCubes-Floyd.png

The number of squares of side $n$ is seen to be $4 n$.

To go from $n$ to $n + 1$, a ring of $4 \paren {n + 1}$ squares of side $n + 1$ is to be added:
 * $4$ for the one at each corner
 * $4 n$ for the ones that abut the sides of the $n + 1$ squares of side $n$.

The length of one side is given by:
 * $S = 2 \paren {1 + 2 + \cdots + n}$

The length of one side is also given by:
 * $S = n \paren {n + 1}$

The area is therefore given in two ways as:


 * $A = 4 \paren {1 + 2 + \cdots + n}^2 = \paren {n \paren {n + 1} }^2$

and also as:

The result follows by equating the expressions for area.