Question

Find the term independent of x in the expansion of

`(x - 2/x^2)^15`

Answer 1

`(x - 2/x^2)^15 = (x + (-2)/x^2)^15` compare with the (x + a)ⁿ 

Here x is x, a is `(-2)/x^2`, n = 15

Let the independent term of x occurs in the general term

`"t"_(r+1) = n"C"_r  x^(n-r) a^r`

`"t"_(r+1) = 15"C"_r  x^(15-r)((-2)/x^2)^r = 15"C"_r  x^(15-r) (-2)^r/(x^2)^r`

`= 15"C"_r  x^(15-r) (-2)^r/(x^(2r))`

`= 15"C"_r  x^(15-r) * x^(-2r) (-2)^r = 15"C"_r  x^(15-r-2r) * (-2)^r`

`= 15"C"_r  x^(15-3r) * (-2)^r`

Independent term occurs only when x power is zero.

15 – 3r = 0

15 = 3r

r = 5

Using r = 5 in (1) we get the independent term

= 15C₅ x⁰ (-2)⁵ [∵ (-2)⁵ = (-1)⁵ 2⁵ = -2⁵]

= -32(15C₅)