Suppose $$$0 \lt = r \lt = 1$$$ and $$$1 \lt = n \lt = 10^9$$$ times we turn $$$r$$$ into $$$3r/(r+2)$$$.
Example: $$$r = 27/47$$$.
$$$n = 1, r = 27/47 = \gt r = 81/121$$$
$$$n = 2, r = 81/121 = \gt r = 243/323$$$
$$$n = 3, r = 243/323 = \gt r = 729/889$$$
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)
Note that the value of $$$1 - 1/r$$$ reduces to $$$2/3$$$ of the original value. So if the answer is $$$r_n$$$ for doing this process $$$n$$$ times, we must have $$$1 - 1/r_n = (2/3)^n (1 - 1/r)$$$. $$$r = 0$$$ can be handled easily.