Question

Which term is independent of x, in the expansion of ${\left(x-\frac{1}{3x^2}\right)}^9?$

Answer 1

$$\begin{gathered} \mathrm{Suppose}{T}_{\mathrm{r}+1}\mathrm{is}\mathrm{the}\mathrm{term}\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{expression}\mathrm{that}\mathrm{is}\mathrm{independent}\mathrm{of}x. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ {T}_{r+1}={}^{9}C_r{x}^{9-r}{\left(\frac{-1}{3x^2}\right)}^r \\ =(-1{)}^r{}^{9}C_r\frac{1}{3^r}{x}^{9-r-2r} \\ \mathrm{For}\mathrm{this}\mathrm{term}\mathrm{to}\mathrm{be}\mathrm{independent}\mathrm{of}x,\mathrm{we}\mathrm{must}\mathrm{have} \\ 9-3r=0 \\ \Rightarrow r=3 \\ \mathrm{Hence},\mathrm{the}\mathrm{required}\mathrm{term}\mathrm{is}\mathrm{the}4\mathrm{th}\mathrm{term}. \end{gathered}$$