Answer
$k+1$
Work Step by Step
If $k$ is an integer, then $\lceil k+\frac{1}{2} \rceil = k+1$.
Let $k=n$, we have $n\lt k+\frac{1}{2}\lt n+1$, thus
$\lceil k+\frac{1}{2} \rceil = n+1=k+1$.
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