Question:
The middle term in the expansion of $\left(\frac{2 x}{3}-\frac{3}{2 x^{2}}\right)^{2 n}$ is
(a) ${ }^{2 n} C_{n}$
(b) $(-1)^{n}{ }^{2 n} C_{n} x^{-n}$
(c) ${ }^{2 n} C_{n} x^{-n}$
(d) none of these
Solution:
(b) $(-1)^{n}{ }^{2 n} C_{n} x^{-n}$
Here, $n$ is even
Middle term in the given expansion $=\left(\frac{2 n}{2}+1\right)$ th $=(n+1)$ th term
$={ }^{2 n} C_{n}\left(\frac{2 x}{3}\right)^{2 n-n}\left(\frac{-3}{2 x^{2}}\right)^{n}$
$=(-1)^{n}{ }^{2 n} C_{n} x^{-n}$
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