Question

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

(b) ${\left(-1\right)}^n{}^{2n}C_n{x}^{-n}$

(c) ${}^{2n}C_n{x}^{-n}$

(d) none of these

Answer 1

(b) ${\left(-1\right)}^n{}^{2n}C_n{x}^{-n}$

$$\begin{gathered} \mathrm{Here},nis\mathrm{even} \\ \mathrm{Middle}\mathrm{term}\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{expansion}=\left(\frac{2n}{2}+1\right)\mathrm{th}=(n+1)\mathrm{th}\mathrm{term} \\ ={}^{2n}C_n{\left(\frac{2x}{3}\right)}^{2n-n}{\left(\frac{-3}{2x^2}\right)}^n \\ =(-1{)}^n{}^{2n}C_n{x}^{-n} \end{gathered}$$