2018 AIME II Problems/Problem 2
Let , , and , and for define recursively to be the remainder when ( ) is divided by . Find .
When given a sequence problem, one good thing to do is to check if the sequence repeats itself or if there is a pattern.
After computing more values of the sequence, it can be observed that the sequence repeats itself every 10 terms starting at .
, , , , , , , , , , , , ,
We can simplify the expression we need to solve to .
Our answer is .
Solution 2 (Overkill)
Notice that the characteristic polynomial of this is
Then since is a root, using Vieta's formula, the other two roots satisfy and .
Let and .
We have so . We found that the three roots of the characteristic polynomial are .
Now we want to express in an explicit form as .
Plugging in we get
And the answer is
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