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Question

# Does the expansion of $\left(2{x}^{2}-\frac{1}{x}\right)$ contain any term involving x9?

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Solution

## Suppose x9 occurs in the given expression at the (r + 1)th term. Then, we have: ${T}_{r+1}={}^{20}C_{r}\left(2{x}^{2}{\right)}^{20-r}{\left(\frac{-1}{x}\right)}^{r}\phantom{\rule{0ex}{0ex}}=\left(-1{\right)}^{r}{}^{20}C_{r}{\left(2\right)}^{20-r}{\left(x\right)}^{40-2r-r}\phantom{\rule{0ex}{0ex}}\mathrm{For}\mathrm{this}\mathrm{term}\mathrm{to}\mathrm{contain}{x}^{9},\mathrm{we}\mathrm{must}\mathrm{have}\phantom{\rule{0ex}{0ex}}40-3r=9\phantom{\rule{0ex}{0ex}}⇒3r=31\phantom{\rule{0ex}{0ex}}⇒r=\frac{31}{3}\phantom{\rule{0ex}{0ex}}\mathrm{It}\mathrm{is}\mathrm{not}\mathrm{possible},\mathrm{as}\mathit{}r\mathrm{is}\mathrm{not}\mathrm{an}\mathrm{integer}.$ Hence, there is no term with x9 in the given expression.

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