# Sum of Sequence of Cubes/Visual Demonstration

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## Theorem

- $\ds \sum_{i \mathop = 1}^n i^3 = \paren {\sum_{i \mathop = 1}^n i}^2 = \frac {n^2 \paren {n + 1}^2} 4$

## Proof

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

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:

\(\ds A\) | \(=\) | \(\ds 4 \times 1^2 + 4 \times 2 \times 2^2 + 4 \times 3 \times 3^2 + \cdots + 4 \times n \times n^2\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 4 \paren {1^3 + 2^2 + 3^3 + \cdots + n^3}\) |

The result follows by equating the expressions for area.

$\blacksquare$

## Historical Note

This visual demonstration of the Sum of Sequence of Cubes was reported by Solomon W. Golomb as having been devised by Warren Lushbaugh.

## Sources

- May 1965: Solomon W. Golomb:
*3121. A Geometric Proof of a Famous Identity*(*Math. Gazette***Vol. 49**,*no. 368*: pp. 198 – 200) www.jstor.org/stable/3612319

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.1$: Mathematical Induction: Exercise $8 \ \text{(b)}$: Figure $5$