Question

Find the term independent of x in the expansion of

`(x^2 - 2/(3x))^9`

Answer 1

Let the independent form of x occurs in the general term, t_r+1 = nC_r x^n-r a^r

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 = 9C_9-6 = 9C₃]