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Nicomachus's Theorem


The nth cubic number n^3 is a sum of n consecutive odd numbers, for example

1^3=1
(1)
2^3=3+5
(2)
3^3=7+9+11
(3)
4^3=13+15+17+19,
(4)

etc. This identity follows from

 sum_(i=1)^n[n(n-1)-1+2i]=n^3.
(5)

It also follows from this fact that

 sum_(k=1)^nk^3=(sum_(k=1)^nk)^2.
(6)

See also

Cubic Number, Odd Number, Odd Number Theorem

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References

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 82, 2003.

Referenced on Wolfram|Alpha

Nicomachus's Theorem

Cite this as:

Weisstein, Eric W. "Nicomachus's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NicomachussTheorem.html

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