Блог пользователя prudent

Reduce n steps of r->3r/(r+2)
Автор prudent, история, 3 года назад, По-английски

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)

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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.

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