Nicomachus's Theorem/Proof 2
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Theorem
\(\ds 1^3\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds 2^3\) | \(=\) | \(\ds 3 + 5\) | ||||||||||||
\(\ds 3^3\) | \(=\) | \(\ds 7 + 9 + 11\) | ||||||||||||
\(\ds 4^3\) | \(=\) | \(\ds 13 + 15 + 17 + 19\) | ||||||||||||
\(\ds \vdots\) | \(\) | \(\ds \) |
In general:
- $\forall n \in \N_{>0}: n^3 = \paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \dotsb + \paren {n^2 + n - 1}$
In particular, the first term for $\paren {n + 1}^3$ is $2$ greater than the last term for $n^3$.
Proof
From the definition:
- $\paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \ldots + \paren {n^2 + n - 1}$
can be written:
- $\paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \ldots + \paren {n^2 - n + 2 n - 1}$
Writing this in sum notation:
\(\ds \) | \(\) | \(\ds \paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \ldots + \paren {n^2 - n + 2 n - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \paren {n^2 - n + 2 k - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n \paren {n^2 - n} + \sum_{k \mathop = 1}^n \paren {2 k - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n^3 - n^2 + n^2\) | Odd Number Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds n^3\) |
$\blacksquare$
Source of Name
This entry was named for Nicomachus of Gerasa.