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Question
Find the term independent of x in the expansion of the expression:
(iii) \[\left( 2 x^2 - \frac{3}{x^3} \right)^{25}\]
Solution
(iii) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
\[\left( 2 x^2 - \frac{3}{x^3} \right)^{25} \]
\[ T_{r + 1} = ^{25}{}{C}_r (2 x^2 )^{25 - r} \left( \frac{- 3}{x^3} \right)^r \]
\[ = ( - 1 )^r {25}{}{C}_r \times 2^{25 - r} \times 3^r x^{50 - 2r - 3r} \]
\[\text{ For this term to be independent of x, we must have} : \]
\[50 - 5r = 0\]
\[ \Rightarrow r = 10\]
\[\text{ Therefore, the required term is the 11th term .} \]
\[\text{ Now, we have } \]
`( - 1 )^{10} "^25C_{10} \times 2^{25 - 10} \times 3^{10} `
` = "^25C_{10 ( 2^{15} \times 3^{10} )`
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