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Question
The middle term in the expansion of \[\left( \frac{2x}{3} - \frac{3}{2 x^2} \right)^{2n}\] is
Options
\[^{2n}{}{C}_n\]
`\left( - 1 \right)^n "^2 n C_n x^{- n}`
\[^{2n}{}{C}_n x^{- n}\]
none of these
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Solution
`\left( - 1 \right)^n "^2 n C_n x^{- n}`
\[\text{ Here, n is even} \]
\[\text{ Middle term in the given expansion } = \left( \frac{2n}{2} + 1 \right)\text{ th = (n + 1)th term } \]
\[ = ^{2n}{}{C}_n \left( \frac{2x}{3} \right)^{2n - n} \left( \frac{- 3}{2 x^2} \right)^n \]
`= ( - 1 )^n "^2nC_n x^{- n}`
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