Sum of Odd Number of Odd Numbers is Odd

Theorem

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

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:

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