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Nicomachus's Theorem

The $n$th Cubic Number $n^3$ is a sum of $n$ consecutive Odd Numbers, for example

\begin{eqnarray*}
1^3 &=& 1\\
2^3 &=& 3+5\\
3^3 &=& 7+9+11\\
4^3 &=& 13+15+17+19,
\end{eqnarray*}



etc. This identity follows from

\begin{displaymath}
\sum_{i=1}^n [n(n-1)-1+2i]=n^3.
\end{displaymath}

It also follows from this fact that

\begin{displaymath}
\sum_{k=1}^n k^3 = \left({\,\sum_{k=1}^n k}\right)^2.
\end{displaymath}

See also Odd Number Theorem




© 1996-9 Eric W. Weisstein
1999-05-25