Question

In the following expansion, find the indicated coefficient.

x^–20 in `(x^3 - 1/(2x^2))^15`

Answer 1

Here, a = x³, b = `(-1)/(2x^2)`, n = 15

We have, t_r+1 = ⁿC_r a^n–r .b^r

= `""^15"C"_"r" (x^3)^(15-"r")*((-1)/(2x^2))^"r"`

= `""^15"C"_"r" x^(45 - 3"r")*((-1)/2)^"r"*x^(-2"r")`

= `""^15"C"_"r"((-1)/2)^"r" x^(45 - 5"r")`

To get the coefficient of x^–20, we must have

x^45–5r = x^–20

∴ 45 – 5r = –20

∴ 5r = 65

∴ r = 13

∴ coefficient of x^–20 = `""^15"C"_13((-1)/2)^13`

= `(15!)/(13!2!) ((-1)/2)^13`

= `(15 xx 14)/(2 xx 1) xx ((-1)/8192)`

= `(-105)/8192`

∴ coefficient of x^–20 is `(-105)/8192`.