2009 AMC 12A Problems/Problem 25
Contents
[hide]Problem
The first two terms of a sequence are and . For ,
What is ?
Solution 1
Consider another sequence such that , and .
The given recurrence becomes
It follows that . Since , all terms in the sequence will be a multiple of .
Now consider another sequence such that , and . The sequence satisfies .
As the number of possible consecutive two terms is finite, we know that the sequence is periodic. Write out the first few terms of the sequence until it starts to repeat.
Note that and . Thus has a period of : .
It follows that and . Thus
Our answer is .
Note
It is not actually difficult to list out the terms until it repeats. You will find that the period is 7 starting from term 2.
See also
2009 AMC 12A (Problems • Answer Key • Resources) | |
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