Question

The middle term in the expansion of ${(\frac{2x}{3}-\frac{3}{2x^2})}^{2n}$ is

• A. ${}^{2n}{C}_n$
• B. $(-1{)}^n{}^{2n}{C}_n{x}^{-n}$
• C. ${}^{2n}{C}_n{x}^{-n}$
• D. None of these

Answer 1

The correct option is B

$(-1{)}^n{}^{2n}{C}_n{x}^{-n}$

$(-1{)}^n{}^{2n}{C}_n{x}^{-n}$

Here, n is even

Middle term in the given expansion $=(\frac{2n}{2}+1)th$ =(n+1)th term

$={}^{2n}{C}_n{\left(\frac{2x}{3}\right)}^{2n-n}{\left(\frac{-3}{2x^2}\right)}^n$

$=(-1{)}^n{}^{2n}{C}_n{x}^{-n}$