User:RandomUndergrad/Sandbox/Generalisation of Nicomachus's Theorem

A Generalization of Nicomachus's Theorem
Nicomachus's Theorem:


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

can be easily be generalized to


 * $\forall n \in \N_{>0}: \forall k \in \R: n^k = \left({n^{k - 1} - n + 1}\right) + \left({n^{k - 1} - n + 3}\right) + \ldots + \left({n^{k - 1} + n - 1}\right)$

by mimicking Nicomachus's Theorem/Proof 2 (basically replacing $n^2$ by $n^{k - 1}$)

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

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

Writing this in sum notation:

However, the property "the first term for $\left({n + 1}\right)^3$ is $2$ greater than the last term for $n^3$" is lost,

all that remains is if $k$ is a strictly positive integer, the summands are consecutive odd integers for each $n$.

This result was found in

Nelsen, R.B. (1993). Proof Without Words: Exercises in Visual Thinking. Math. Assoc. of America, 1993. ISBN 0-88385-700-6, pp.93,

with the title "$k^\text{th}$ Powers as Sums of Consecutive Numbers", where a visual representation of the result is shown, for $k=2,3\dots$