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Question
Find the term independent of x, x ≠ 0, in the expansion of `((3x^2)/2 - 1/(3x))^15`
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Solution
General Term `"T"_(r + 1) = ""^n"C"_r x^(n - r) y^r`
= `""^15"C"_r ((3x^2)/2)^(15 - r) (- 1/(3x))^r`
= `""^15"C"_r (3/2)^(15 - r) * (x)^(30 - 2r) * (- 1/3)^r * 1/x^r`
= `""^15"C"_r (3/2)^(15 - r) * (x)^(30 - 2r - r) (-1)^r * 1/(3)^r`
= `""^15"C"_r (3/2)^(15 - r) * x^(30 - 3r) (-1)^r * 1/(3)^r`
For getting the term independent of x
30 – 3r = 0
⇒ r = 10
On putting the value of r in the above expression, we get
= `""^15"C"_10 (3/2)^(15 - 10) (-1)^10 * 1/(3)^10`
= `""^15"C"_10 (3)^5/(2)^5 * 1/(3)^10`
= `""^15"C"_10 * 1/((2)^5 * (3)^5)`
= `""^15"C"_10 (1/6)^5`
Hence, the required term = `""^15"C"_10 (1/16)^5`
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