Question

Middle term of the expression (x+y)^n

Answer 1

If n is even, the middle term is $\binom{n}{n/2} x^{n/2} y^{n/2}$. If n is odd, the middle terms are $\binom{n}{(n-1)/2} x^{(n+1)/2} y^{(n-1)/2}$ and $\binom{n}{(n+1)/2} x^{(n-1)/2} y^{(n+1)/2}$.

Answer 2

The expansion of $(x+y)^n$ has $(n+1)$ terms. The determination of the middle term(s) depends on whether $n$ is an even or an odd integer.

The general term in the expansion of $(x+y)^n$ is given by the formula: $T_{r+1} = \binom{n}{r} x^{n-r} y^r$

Case 1: $n$ is an even integer. If $n$ is even, then $n+1$ (the total number of terms) is an odd integer. In this case, there is exactly one middle term. The position of the middle term is $\left(\frac{n}{2} + 1\right)$. To find this term using the general term formula, we set $r+1 = \frac{n}{2} + 1$, which implies $r = \frac{n}{2}$. Substituting $r=\frac{n}{2}$ into the general term formula: Middle Term $= T_{\frac{n}{2}+1} = \binom{n}{n/2} x^{n-n/2} y^{n/2} = \binom{n}{n/2} x^{n/2} y^{n/2}$.

Case 2: $n$ is an odd integer. If $n$ is odd, then $n+1$ (the total number of terms) is an even integer. In this case, there are two middle terms. The positions of the two middle terms are $\left(\frac{n+1}{2}\right)$ and $\left(\frac{n+1}{2} + 1\right)$.

• First Middle Term: The position of the first middle term is $\left(\frac{n+1}{2}\right)$. To find this term, we set $r+1 = \frac{n+1}{2}$, which implies $r = \frac{n+1}{2} - 1 = \frac{n-1}{2}$. Substituting $r=\frac{n-1}{2}$ into the general term formula: First Middle Term $= T_{\frac{n-1}{2}+1} = T_{\frac{n+1}{2}} = \binom{n}{(n-1)/2} x^{n-(n-1)/2} y^{(n-1)/2} = \binom{n}{(n-1)/2} x^{(n+1)/2} y^{(n-1)/2}$.

• Second Middle Term: The position of the second middle term is $\left(\frac{n+1}{2} + 1\right)$. To find this term, we set $r+1 = \frac{n+1}{2} + 1$, which implies $r = \frac{n+1}{2}$. Substituting $r=\frac{n+1}{2}$ into the general term formula: Second Middle Term $= T_{\frac{n+1}{2}+1} = \binom{n}{(n+1)/2} x^{n-(n+1)/2} y^{(n+1)/2} = \binom{n}{(n+1)/2} x^{(n-1)/2} y^{(n+1)/2}$.

Summary:

• If $n$ is even, the middle term is $T_{\frac{n}{2}+1} = \binom{n}{n/2} x^{n/2} y^{n/2}$.
• If $n$ is odd, the two middle terms are $T_{\frac{n+1}{2}} = \binom{n}{(n-1)/2} x^{(n+1)/2} y^{(n-1)/2}$ and $T_{\frac{n+1}{2}+1} = \binom{n}{(n+1)/2} x^{(n-1)/2} y^{(n+1)/2}$.