Find the middle term of $(1+x)^{2n}$ in its simplest form.

Question

Find the middle term of $(1+x)^{2n}$ in its simplest form.

There are a total of $2n+1$ terms. The middle term is $\dfrac{2n+1}{2}$.

Hence $^{2n}C_{\frac{2n+1}{2}}x^{\frac{2n+1}{2}}$

But I feel I got the index of the middle term wrong. Also I'm not sure how to incorporate consideration of whether $n$ is odd or even.

Answer 1

In the sequence $$\{0,1,2,3,4,5,6\},$$ what is the middle term? It's $3$.

Similarly, the binomial expansion $(1+x)^6$ is $$1x^0 + 6x^1 + 15x^2 + 20x^3 +15x^4 + 6x^5 + 1x^6.$$ The middle term is $$20x^3.$$ What was the value of $n$ in the above example? It was $3$, since $2(3) = 6$. Do you now see how this works in the case of general $n$, and why the middle term of $(1+x)^{2n}$ has the form $$Cx^n$$ for some coefficient $C$?

Answer 2

Since there are $2n+1$ terms, the middlemost term would be the $(n+1)$th term.

For the $(n+1)$th term, you would require putting $r=n$ and not $n+1$ ($r$ varies from $0$ to $2n$ thereby giving $2n+1$ terms).

So the coefficient of the middlemost term will be $\binom{2n}{n}x^n$.