Question

The coefficient of $x^7$ in $(\frac{x^2}{2}-\frac{2}{x}{)}^9$ is

• A. $-56$
• B. $14$
• C. $-14$
• D. $0$

Answer 1

The correct option is D $0$

Consider the following
$T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r$
Applying to the above question we get
$T_{r+1}=(-1){}^r{}^9{C}_r\left(\frac{x^2}{2}\right){}^{9-r}\left(2x^{-1}\right){}^r$
$T_{r+1}=(-1){}^r{}^9{C}_r{x}^{18-3r}{2}^{2r-9}$
Therefore $18-3r=7$ for the coefficient of $x^7$
$18-3r=7$
$11=3r$
$r=\frac{11}{3}$ but $r$ has to be a whole number.
Hence the coefficient of $x^7$ is zero that is there is no $x^7$ term in the expansion