2007 iTest Problems/Problem 47
Problem
Let and be sequences defined as follows: ,
Let be the largest integer that satisfies all of the following conditions: , for some positive integer ; , for some positive integer ; and . Find the remainder when is divided by .
Solution
First, we solve these linear recurrences. The characteristic polynomial of is which has roots -1 and 2. Then with the given values, where and are the solutions to the system Solving, we find and , so . Similarly, .
We can ignore the terms because they will be inconsequential compared to the terms, so define and . Note that for some and , so that we can place at the average of and at least. Therefore and will have to be somewhat close to each other, so we examine the equation , or . Solving for results in , and because and are integers, .
Using our approximations in our inequality, we find , which simplifies to . Bashing or calculator use results in , so and . Note that , so the largest possible will be . The requested answer is .
~clarkculus
See Also
2007 iTest (Problems, Answer Key) | ||
Preceded by: Problem 46 |
Followed by: Problem 48 | |
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