The Middle Term(s)

Why the Middle Term Depends on Parity

The expansion of $(a+b)^n$ always has $n+1$ terms (Section 2, Pattern (i)). Whether this total, $n+1$, is odd or even depends entirely on whether $n$ itself is even or odd, and that parity decides whether the expansion has a single middle term or two middle terms.

Case 1: $n$ even. Then $n+1$ is odd, so there is exactly one term exactly in the middle of the list of $n+1$ terms. Its position, counting from the $1$st term, is $\dfrac{n}{2}+1$. Using the general term with $p=\dfrac{n}{2}+1$, i.e. $r=\dfrac{n}{2}$, the single middle term is

Case 2: $n$ odd. Then $n+1$ is even, so there is no single middle position -- instead there are exactly two middle terms, positioned symmetrically on either side of the centre. Their positions are $\dfrac{n+1}{2}$ and $\dfrac{n+1}{2}+1=\dfrac{n+3}{2}$, giving

A Quick Way to Remember It

Rather than memorising the two boxed formulas above, it is enough to remember the rule in words: count the total number of terms ($n+1$); if that count is odd, there is one middle term at position (total$+1)/2$; if that count is even, there are two middle terms at positions total$/2$ and total$/2+1$. This single rule, applied to the term count $n+1$, reproduces both cases without needing to memorise separate formulas for "$n$ even" and "$n$ odd" -- and it is exactly the same rule used to find the median position of any ordered list of $n+1$ items. …