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Question
The middle term in the expansion of \[\left( \frac{2 x^2}{3} + \frac{3}{2 x^2} \right)^{10}\] is
Options
251
252
250
none of these
Solution
252
\[\text{ Here, n, i . e . , 10, is an even number } . \]
\[ \therefore \text{ Middle term } = \left( \frac{10}{2} + 1 \right)\text{ th term = 6th term} \]
\[\text{ Thus, we have: } \]
\[ T_6 = T_{5 + 1} \]
\[ = ^{10}{}{C}_5 \left( \frac{2 x^2}{3} \right)^{10 - 5} \left( \frac{3}{2 x^2} \right)^5 \]
\[ = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2} \times \frac{2^5}{3^5} \times \frac{3^5}{2^5}\]
\[ = 252\]
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