Where am I going wrong in simplifying this expression with gamma functions?

Question

I am getting the following, using the property $\Gamma(z) = z\Gamma(z-1)$

$\frac{1}{2}\frac{\Gamma(\frac{v}{2}-1)}{\Gamma(\frac{v}{2})} = \frac{1}{2}\frac{\Gamma(\frac{v}{2}-1)}{\frac{v}{2}\times\Gamma(\frac{v}{2}-1)} = \frac{1}{v}$

But the answer is supposed to be $\frac{1}{v-2}$

Answer 1

The correct recurrence for the gamma function is $\Gamma(x+1)=x\Gamma(x)$ or $\Gamma(x)=(x-1)\Gamma(x-1)$ so the $\frac v2$ in the denominator should be $\frac v2-1$