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Question
Find the term independent of x in the expansion of `(3x - 2/x^2)^15`
Solution
Given expression is `(3x - 2/x^2)^15`
General term `"T"_(r + 1) = ""^n"C"_r x^(n - r) y^r`
= `""^15"C"_r (3x)^(15 - r) (- 2/x^2)^r`
= `""^15"C"_r (3)^(15 - r) * x^(15 - r) (-2)^r * 1/x^(2r)`
= `""^15"C"_r (3)^(15 - r) * x^(15 - r - 2r) * (-2)^r`
= `""^15"C"_r (3)^(15 - r) * x^(15 - 3r) (-2)^r`
For getting a term independent of x
Put 15 – 3r = 0
⇒ r = 5
∴ The required term is `""^15"C"_5 (3)^(15 - 5) (-2)^5`
= `- ""^15"C"_3 (3)^10 (2)^5`
= `-(15 xx 14 xx 13 xx 12 xx 11)/(5 xx 4 xx 3 xx 2 xx 1) * (3)^10 (2)^5`
= `-7 xx 13 xx 3 xx 11 * 3(3)^10 (2)^5`
= – 3003 (3)10 (2)5
Hence, the required term = –3003 (3)10 (2)5
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