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 , we get the series , which seem to always be a negative power of away from . We can test this out by setting to .
Now,
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
See Also
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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