Here's a quick and simple one.

The arithmetic average of 2kn^2 and 2k(n+1)^2 is always k greater than their geometric average for positive integers k and n.

There wasn't a uniform way to find those 2 numbers- I had to do it by finding a bunch that worked and looking for a pattern.

Note: The arithmetic average of 2 numbers a and b is (a+b)/2, and their geometric average is the Sqrt(a*b).