Question

In the following expansion, find the indicated coefficient.

x⁸ in `(2x^5 - 5/x^3)^8`

Answer 1

Let t_r+1 contains x⁸ in the expansion of `(2x^5 - 5/x^3)^8`

We know that, in the expansion of (a + b)ⁿ,

t_r+1 = ⁿC_r a^n–r b^r 

Here a = 2x⁵, b = `-5/x^3`, n = 8

∴ t_r+1 = `""^8"C"_"r" (2x^5)^(8 - "r") (-5/x^3)^"r"`

= `""^8"C"_"r".2^(8-"r").x^(40-5"r").(-5)^"r"/x^(3"r")` 

= ⁸C_r .2^8–r. (–5)^r. x^40–8r 

But t_r+1 contains x⁸

∴ power of x = 8

∴ 40 – 8r = 8

∴  r = 4

∴ coefficient of x⁸ = ⁸C₄ . 2^8–4 . (–5)⁴ 

= `(8 xx 7 xx 6 xx 5)/(1 xx 2 xx 3 xx 4) xx 2^4 xx 625`

= 70 × 16 × 625

= 700000