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Question
Find the middle terms(s) in the expansion of:
(viii) \[\left( 2ax - \frac{b}{x^2} \right)^{12}\]
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Solution
\[\left( 2ax - \frac{b}{x^2} \right)^{12} \]
\[\text{ Here, n is an even number .} \]
\[ \therefore \text{ Middle term } = \left( \frac{12}{2} + 1 \right)^{th} = 7^{th} \text{ term } \]
\[\text{ Now, we have } \]
\[ T_7 = T_{6 + 1} \]
\[ = ^{12}{}{C}_6 \left( 2ax \right)^{12 - 6} \left( \frac{- b}{x^2} \right)^6 \]
\[ = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \times \left( \frac{2ab}{x} \right)^6 \]
\[ = \frac{59136 a^6 b^6}{x^6}\]
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