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Question
Find the term independent of x in the expansion of
`(x^2 - 2/(3x))^9`
Solution
Let the independent form of x occurs in the general term, tr+1 = nCr xn-r ar
Here x is x2, a is `(-2)/(3x)` and n = 9
∴ `"t"_(r+1) = 9"C"_"r" (x^2)^(9-r) ((-2)/(3x))^r = 9"C"_r x^(2(9-r)) ((-2)^r/(3^rx^r))`
`= 9"C"_r x^(18-2r) * x^(-r) (-2)^r/3^r`
`= 9"C"_r x^(18-2r-r) (-2)^r/3^r = 9"C"_r x^(18-3r) (-2)^r/3^r`
Independent term occurs only when x power is zero.
18 – 3r = 0
⇒ 18 = 3r
⇒ r = 6
Put r = 6 in (1) we get the independent term as 9C6 x0 `(-2)^6/3^6`
`= 9"C"_3 (2/3)^6` ..[∵ 9C6 = 9C9-6 = 9C3]
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