What is the coefficient of a x10 b x11 in the expansion of 2

What is the coefficient of

(a) x¹⁰

(b) x¹¹

in the expansion of (2x 1/x²)¹⁶?

Solution

the general term of ( x + y )ⁿ

T_r+1 = ⁿC_r * x(^n-r) * y^r

so, given question ( 2x - 1/x²)¹⁶

T_r+1 = ¹⁶C_r * (2x)^16-r * (-1/x²)^r

T_r+1 = ¹⁶C_r * 2^16-r * x^16-r * (-1)^r/x^2r

T_r+1 = (-1)^r * ¹⁶C_r * 2^16-r * x^16-r-2r

( A) the coefficient of x¹⁰,

16-r-2r = 10

r = 2

hence,the coefficient of x¹⁰= (-1)² * ¹⁶C₂ * 2^16-2

= ¹⁶C₂ * 2¹⁴

(B)

the coefficient of x¹¹,

16-r-2r = 11

r = 5/3

which is not possible.

so,there is no term like x¹¹