The middle term in the expansion of

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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