Suppose $$$0 <= r <= 1$$$ and $$$1 <= n <= 10^9$$$ times we turn $$$r$$$ into $$$3r/(r+2)$$$.
Example: $$$r = 27/47$$$.
$$$n = 1, r = 27/47 => r = 81/121$$$
$$$n = 2, r = 81/121 => r = 243/323$$$
$$$n = 3, r = 243/323 => r = 729/889$$$
Is this process reducible to a formula?
P.S. This problem is not from a contest, but a subproblem of an assignment on MIT OpenCourseWare. (5th problem, solving for an arbitrary number of 1's)