2019 AMC 10 B

We first prove that  for all , by induction. Observe that

so (since is clearly positive for all , from the initial definition), if and only if .

We similarly prove that  is decreasing, since

Now we need to estimate the value of  , which we can do using the rearranged equation

Since   is decreasing, is clearly also decreasing, so we

have and

This becomes   The problem thus reduces to finding the least value of  such that and

Taking logarithms, we get   and , i.e. and .

As approximations, we can use , , and . These allow us to estimate that   which gives the answer as .

The condition where   gives the motivation to make a substitution to change the equilibrium from   to . We can substitute to achieve that.

Now, we need to find the smallest value of  such that given that and the recursion  .

Using wishful thinking, we can simplify the recursion as follows:

The recursion looks like a geometric sequence with the ratio changing slightly after each term. Notice from the recursion that the  sequence is strictly decreasing, so all the terms after  will be less than . Also, notice that all the terms in sequence will be positive. Both of these can be proven by induction. With both of those observations in mind, . Combining this with the fact that the recursion resembles a geometric sequence, we conclude that . is approximately equal to and the ranges that the answer choices give us are generous, so we should use either or to find a rough estimate for .  is , while   is close to because is  , which is close to .

Therefore, we can estimate that . Putting both sides to the power, we get   But , so and therefore, . This tells us that is somewhere around , so our answer is .