Question:
Does the expansion of $\left(2 x^{2}-\frac{1}{x}\right)$ contain any term involving $x^{9} ?$
Solution:
Suppose $x^{9}$ occurs in the given expression at the $(r+1)$ th term.
Then, we have:
$T_{r+1}={ }^{20} C_{r}\left(2 x^{2}\right)^{20-r}\left(\frac{-1}{x}\right)^{r}$
$=(-1)^{r}{ }^{20} C_{r}(2)^{20-r}(x)^{40-2 r-r}$
For this term to contain $x^{9}$, we must have
$40-3 r=9$
$\Rightarrow 3 r=31$
$\Rightarrow r=\frac{31}{3}$
It is not possible, as $r$ is not an integer.
Hence, there is no term with $x^{9}$ in the given expression.
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