Nicomachus's Theorem/Proof 2

Theorem
In general:
 * $\forall n \in \N_{>0}: n^3 = \left({n^2 - n + 1}\right) + \left({n^2 - n + 3}\right) + \ldots + \left({n^2 + n - 1}\right)$

In particular, the first term for $\left({n + 1}\right)^3$ is $2$ greater than the last term for $n^3$.

Proof
From the definition:
 * $\left({n^2 - n + 1}\right) + \left({n^2 - n + 3}\right) + \ldots + \left({n^2 + n - 1}\right)$

can be written:
 * $\left({n^2 - n + 1}\right) + \left({n^2 - n + 3}\right) + \ldots + \left({n^2 - n + 2 n - 1}\right)$

Writing this in sum notation: