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Question
Find the coefficient of:
(iii) \[x^{- 15}\] in the expansion of \[\left( 3 x^2 - \frac{a}{3 x^3} \right)^{10}\]
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Solution
(iii) Suppose x−15 occurs at the (r + 1)th term in the given expression.
Then, we have:
\[\Rightarrow T_{r + 1} = ( - 1 )^r {10}{}{C}_r \left( 3^{10 - r - r} \right)\left( x^{20 - 2r - 3r} \right)\left( a^r \right)\]
\[\text{ For this term to contain } x^{- 15} ,\text{ we must have} : \]
\[20 - 5r = - 15\]
\[ \Rightarrow 5r = 20 + 15\]
\[ \Rightarrow r = 7\]
\[ \therefore \text{ Coefficient of } x^{- 15} = ( - 1 )^7 {10}{}{C}_7 3^{10 - 14} \left( a^7 \right) = - \frac{10 \times 9 \times 8}{3 \times 2 \times 9 \times 9} a^7 = - \frac{40}{27} a^7\]
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