2021 Fall AMC 12B Problems/Problem 18
Problem
Set , and for
let
be determined by the recurrence
This sequence tends to a limit; call it . What is the least value of
such that
Solution
If we list out the first few values of k, we get the series , which seem to always be a negative power of 2 away from
. We can test this out by setting
to
.
Now,
This means that .
We see that seems to always be
above a power of
. We can prove this using induction.
Claim:
Base case:
Induction:
It follows that , and
. Therefore, the least value of
would be
.
-ConcaveTriangle