JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Middle Term

The middle term depends upon the value of n.

(1) When n is even, then total number of terms in the expansion of \[{{(x+y)}^{n}}\] is \[n+1\] (odd). So there is only one middle term i.e., \[{{\left( \frac{n}{2}+1 \right)}^{\text{th}}}\] term is the middle term. \[{{T}_{\left[ \frac{n}{2}+1 \right]}}{{=}^{n}}{{C}_{n/2}}{{x}^{n/2}}{{y}^{n/2}}\]

(2) When n is odd, then total number of terms in the expansion of \[{{(x+y)}^{n}}\] is \[n+1\] (even). So, there are two middle terms i.e.,\[{{\left( \frac{n+1}{2} \right)}^{\text{th}}}\] and \[{{\left( \frac{n+3}{2} \right)}^{\text{th}}}\] are two middle terms.

 \[{{T}_{\left( \frac{n+1}{2} \right)}}{{=}^{n}}{{C}_{\frac{n-1}{2}}}{{x}^{\frac{n+1}{2}}}{{y}^{\frac{n-1}{2}}}\]  and \[{{T}_{\left( \frac{n+3}{2} \right)}}{{=}^{n}}{{C}_{\frac{n+1}{2}}}{{x}^{\frac{n-1}{2}}}{{y}^{\frac{n+1}{2}}}\]

• When there are two middle terms in the expansion then their binomial coefficients are equal.

• Binomial coefficient of middle term is the greatest binomial coefficient.