Middle Term(s) in the Expansion of (a+b)^n

How many terms, and where is the middle? The expansion of $(a+b)^n$ always has $n+1$ terms (Remark 1 of Section 4.2). Whether that count is odd or even decides whether there is a single middle term or a pair of them.

• If $n$ is even, $n+1$ is odd, so there is exactly one middle term: the $\left(\dfrac n2+1\right)$th term.
• If $n$ is odd, $n+1$ is even, so there are exactly two middle terms: the $\left(\dfrac{n+1}2\right)$th term and the $\left(\dfrac{n+3}2\right)$th term (these are consecutive terms).

In either case, once the position of the middle term(s) is known, its value is found the same way as any other term — by substituting the matching value of $r$ into the general-term formula $t_{r+1}={}^nC_ra^{n-r}b^r$ from Section 4.3.

Solved Example 1. Find the middle term(s) in the expansion of $\left(x^2+\dfrac2x\right)^8$. Solution. Here $a=x^2,b=2/x,n=8$. Since $n=8$ is even, $\dfrac n2+1=5$, so the fifth term is the only middle term. For $t_5$, $r=4$: $t_5={}^8C_4(x^2)^{8-4}\left(\dfrac2x\right)^4={}^8C_4x^8\cdot\dfrac{16}{x^4}=70(x^8)\dfrac{16}{x^4}=1120x^4$. …