Binomial Theorem — Middle Term(s) and Applications

Locating the middle term. The expansion of $(x+y)^n$ has $n+1$ terms.

• If $n$ is even, there is a single middle term, the $\left(\dfrac{n}{2}+1\right)$th term.
• If $n$ is odd, there are two middle terms, the $\left(\dfrac{n+1}{2}\right)$th and $\left(\dfrac{n+3}{2}\right)$th terms.

In either case, the middle term (or terms) is found by substituting the correct value of $r$ into the general term formula $T_{r+1} = \,^{n}C_{r}\,x^{n-r}y^{r}$ from the previous section.

Application — approximation. When $y$ is small compared to $x$ (typically $x=1$ and $y$ a small decimal correction), the terms of $(1+y)^n$ shrink rapidly, since each term carries a higher power of $y$ than the last. Keeping only the first few terms — often just up to $y^2$ — gives a good decimal approximation without needing to compute the full power directly:

This technique is genuinely useful in business calculations — for instance, approximating a compound-growth factor $(1+r)^n$ for a small interest or growth rate $r$ without a calculator, by keeping only the first two or three terms. …