The 
th
cubic number 
is a sum of 
consecutive odd numbers, for
example
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
etc. This identity follows from
![sum_(i=1)^n[n(n-1)-1+2i]=n^3.](/images/equations/NicomachussTheorem/NumberedEquation1.svg) |
(5)
|
It also follows from this fact that
 |
(6)
|
Nicomachus's Theorem
See also
Cubic Number, Odd Number, Odd Number TheoremExplore with Wolfram|Alpha
References
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 82, 2003.Referenced on Wolfram|Alpha
Nicomachus's TheoremCite this as:
Weisstein, Eric W. "Nicomachus's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NicomachussTheorem.html