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Question
Find the middle terms(s) in the expansion of:
(vi) \[\left( \frac{x}{3} + 9y \right)^{10}\]
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Solution
\[\left( \frac{x}{3} + 9y \right)^{10} \]
\[\text{ Here, n is an even number } . \]
\[\text{ Therefore, the middle term is } \left( \frac{10}{2} + 1 \right)\text{ th, i . e . , 6th term . } \]
\[\text{ Now, we have } \]
\[ T_6 = T_{5 + 1} \]
\[ =^{10}{}{C}_5 \left( \frac{x}{3} \right)^{10 - 5} (9y )^5 \]
\[ = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2} \times \frac{1}{3^5} \times 9^5 \times x^5 y^5 \]
\[ = 61236 x^5 y^5\]
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