Sum of Odd Number of Odd Numbers is Odd

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Theorem

In the words of Euclid:

If as many odd numbers as we please be added together, and their multitude is odd, the whole is odd.

(The Elements: Book $\text{IX}$: Proposition $23$)


Proof

Let $S = \left\{{r_1, r_2, \ldots, r_n}\right\}$ be a set of $n$ odd numbers, where $n = 2 m + 1$.

By definition of odd number, this can be expressed as:

$S = \left\{{2 s_1 + 1, 2 s_2 + 1, \ldots, 2 s_n + 1}\right\}$

where:

$\forall k \in \left[{1 \,.\,.\, n}\right]: r_k = 2 s_k + 1$

Then:

\(\ds \sum_{k \mathop = 1}^n r_k\) \(=\) \(\ds \sum_{k \mathop = 1}^{2 m + 1} \left({2 s_k + 1}\right)\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^{2 m} \left({2 s_k + 1}\right) + 2 s_n + 1\) extracting the $n$th element from the summation
\(\ds \) \(=\) \(\ds 2 q + 2 s_n + 1\) where $q \in \N$: Sum of Even Number of Odd Numbers is Even
\(\ds \) \(=\) \(\ds 2 \left({q + s_n}\right) + 1\)

Thus, by definition, $\displaystyle \sum_{k \mathop = 1}^n r_k$ is odd.

$\blacksquare$


Historical Note

This proof is Proposition $23$ of Book $\text{IX}$ of Euclid's The Elements.


Sources