Answer
\begin{equation}
N_{m}=\frac{n-1}{2}+1=\frac{n-1+2}{2}=\frac{n+1}{2}
\end{equation}
Work Step by Step
We suppose we have an odd number of items, like $15 .$ Our task is to
find the position for the middle item. Here's how we can find the position for
a 15 -element list:
$15=14+1=2 \cdot 7+1=7+1+7$
Here we can see that we have one single element in the middle because it is
an odd number and odd numbers can be written in the form $2 \cdot m+1 .$ It
is obvious that the middle element is the $8^{t h}$ element which has 7 elements
before and 7 after it.
We can do the same to a couple of other odd numbers:
$\quad \quad \bullet 23=2 \cdot 11+1,$ with $12^{t h}$ as the middle element
$\quad \quad \bullet 75=2 \cdot 37+1,$ with $38^{t h}$ as the middle element
The number of comparisons required to find the middle item for each of the
above lists is equal to the middle's position: $8,12,38 .$
We can write an expression to find this middle item $N_{m}$ in a $n$ -element
list, where $n$ is odd:
$N_{m}=\frac{n-1}{2}+1=\frac{n-1+2}{2}=\frac{n+1}{2}$
