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Question
Find the term independent of x in the expansion of the expression:
(ix) \[\left( \sqrt[3]{x} + \frac{1}{2 \sqrt[3]{x}} \right)^{18} , x > 0\]
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Solution
(ix) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
\[\left( \sqrt[3]{x} + \frac{1}{2\sqrt[3]{x}} \right)^{18} \]
\[ T_{r + 1} =^{18}{}{C}_r ( x^{1/3} )^{18 - r} \left( \frac{1}{2 x^{1/3}} \right)^r \]
\[ =^{18}{}{C}_r \times \frac{1}{2^r} x^\frac{18 - r}{3} - \frac{r}{3} \]
\[\text{ For this term to be independent of r, we must have } \]
\[\frac{18 - r}{3} - \frac{r}{3} = 0\]
\[ \Rightarrow 18 - 2r = 0\]
\[ \Rightarrow r = 9\]
\[\text{ The term is } \]
\[^{18}{}{C}_9 \times \frac{1}{2^9}\]
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