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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:
\[T_{r + 1} =^{20}{}{C}_r (2 x^2 )^{20 - r} \left( \frac{- 1}{x} \right)^r \]
\[ = ( - 1 )^r {20}{}{C}_r \left( 2 \right)^{20 - r} \left( x \right)^{40 - 2r - r} \]
\[\text{ For this term to contain } x^9 , \text{ we must have} \]
\[40 - 3r = 9\]
\[ \Rightarrow 3r = 31\]
\[ \Rightarrow r = \frac{31}{3} \]
\[\text{ It is not possible, as r is not an integer } .\]
Hence, there is no term with x9 in the given expression.
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