A Visual Proof of Nicomachus's Theorem
A Visual Proof of Nicomachus's Theorem
Nicomachus's theorem states that ++…+=, where is a positive integer. In words, the sum of the cubes from 1 to is equal to the square of the sum from 1 to .
3
1
3
2
3
n
2
(1+2+…+n)
n
n
n
For a visual proof, calculate the total area in the figure in two different ways: First, count the unit squares from the center to an edge to get , so that the total area is . Second, consider that each square ring consists of squares of side , with area .
1+2+3+...+n
4
2
(1+2+...+n)
4k
k
4
3
k