2021 Fall AMC 12B Problems/Problem 18
Contents
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 1
If we list out the first few values of , we get the series , which seem to always be a negative power of away from . We can test this out by setting to .
Now, we get This means that this series approaches , as the second term is decreasing. In addition, we find that .
We see that seems to always be above a power of . We can prove this using induction.
Claim
Base case
We have , which is true.
Induction Step
Assuming that the claim is true, we have .
It follows that , and . Therefore, the least value of would be .
~ConcaveTriangle
Solution 2
Note that all terms of the sequence lie in the interval strictly increasing.
Since the sequence tends to the limit we set The given equation becomes from which
See Also
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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All AMC 12 Problems and Solutions |
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