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Question
Find the coefficient of:
(i) x10 in the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)^{20}\]
Solution
(i) Suppose x10 occurs in the (r + 1)th term in the given expression.
Then, we have: \[T_{r + 1} =^{n}{}{C}_r x^{n - r} a^r\]
Here,
\[T_{r + 1} = ^{20}{}{C}_r (2 x^2 )^{20 - r} \left( \frac{- 1}{x} \right)^r \]
`=(-1)^r "^20C_r (2^(20-r) ) ( x^(40-2r-r) )`
\[\text{ For this term to contain } x^{10} , \text{ we must have:} \]
\[40 - 3r = 10\]
\[ \Rightarrow 3r = 30\]
\[ \Rightarrow r = 10\]
` therefore "Coefficient of" x^10 = (-1)^10 ^20 C_10 (2^(20-10))="^20C_10 (2^10)`
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