Nicomachus’ theorem

Theorem (Nicomachus). The sum of the cubes of the first n integers is equal to the square of the nth triangular numberMathworldPlanetmath. To put it algebraically,


There are several formulas for the triangular numbers. Gauss figured out that to compute


one can, instead of summing the numbers one by one, pair up the numbers thus: 1+n, 2+(n-1), 3+(n-2), etc., and each of these sums has the same result, namely, n+1. Since there are n of these sums, carrying this all the way through to the end, we are in effect squaring n+1, which is (n+1)2=(n+1)(n+1)=n2+n. But this is redundant, since it includes both 1+n and n+1, both 2+(n-1) and (n-1)+2, etc., in effect, each of these twice. Therefore,


As Sir Charles Wheatstone proved, we can rewrite i3 as


That sum can always be rewritten as a sum of odd terms, namely


Thus, the sum of the first n cubes is in fact also


The sum of the first n-1 odd numbersMathworldPlanetmathPlanetmath is n2, and therefore


as the theorem states. ∎

For example, the sum of the first four cubes is 1 + 9 + 27 + 64 = 100. This is also equal to 1 + 3 + 5 + 7 + 9 + 11 + 13 + 17 + 19 = 100. The square root of 100 is 10, the fourth triangular number, and indeed 10 = 1 + 2 + 3 + 4.

Title Nicomachus’ theorem
Canonical name NicomachusTheorem
Date of creation 2013-03-22 18:07:12
Last modified on 2013-03-22 18:07:12
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 7
Author PrimeFan (13766)
Entry type Theorem
Classification msc 11A25
Related topic CubeOfAnInteger