Hi, I am working on a project dealing with n-spheres.

I have an even equation

Vn=(1/n!)*(pi^n)*(R^2n)

and an odd equation

Vn=((2^(2n+1)n!)/(2n+1)!)*(pi^n)*R^2n+1

How can I prove these? I have no idea how to arrive at them. We have one general formula,

Vn(R)=2(integral from 0 to pi/2)Vn-1(Rcos(theta))Rcos(theta)d(theta)