Python 3, 65 55 bytes

Thanks to mathmandan for pointing out a shorter version using the lambda operator.

f=lambda a,b,n:f((a+b)/2,(a*b)**.5,n-1)if n else(a,b)

My original version:

def f(a,b,n):
 if n:f((a+b)/2,(a*b)**.5,n-1)
 else:print(a,b)

To my chagrin, a recursive function (a la the JavaScript and C++ answers) was shorter than a simple for loop.