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 $5$