2019 AMC 12A Problems/Problem 23
Contents
Problem
Define binary operations and by for all real numbers and for which these expressions are defined. The sequence is defined recursively by and for all integers . To the nearest integer, what is ?
Solution 1
By definition, the recursion becomes . By the change of base formula, this reduces to . Thus, we have . Thus, for each positive integer , the value of must be some constant value .
We now compute from . It is given that , so .
Now, we must have . Changing bases to , this becomes , so , where the last equality comes from the logarithmic chain rule. We conclude that , or choice .
Solution 2
Using the recursive definition, or where and . Using logarithm rules, we can remove the exponent of the 3 so that . Therefore, , which is .
We claim that for all . We can prove this through induction.
See Also
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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