If $n$ is odd, then $n/2 + 1/2$ is always even? If $n$ is odd, prove that $n/2 + 1/2$ is even.

Context: I'm a Statistician and the term $n/2 + 1/2$ showed up in the index of a summation when deriving the pdf of some Order Statistic:
$$
\sum_{j = (n+1)/2}^{n}...
$$
I realized that $n/2 + 1/2$ is always even if $n$ is odd, but I couldn't prove the result to myself (well, I have no training in Number Theory).
What I've tried:
Suppose $n$ is odd. Then $n + 1$ is even (by the successor function?). Then $n + 1 = 2k$ for $k \in \mathbb{N} \Rightarrow (n+1)/2 = k $. But this doesn't show that $k$ is even.
No.
Consider $n=1$.
I add this sentence because my answer is too short.
Odd numbers are $2$ units apart, so that the half odd numbers are $1$ unit apart and induce both parities.