Difference between revisions of "2009 AMC 12A Problems/Problem 25"
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==Note== | ==Note== | ||
− | It is not actually difficult to list out the terms until it repeats. You will find that the period is | + | It is not actually difficult to list out the terms until it repeats. You will find that the period is 24 starting from term 2. |
== See also == | == See also == |
Revision as of 22:57, 8 December 2018
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 .
Solution 2:
You didn't have to do what the user above did, just try some values and then you will find it repeats the answer is A
Note
It is not actually difficult to list out the terms until it repeats. You will find that the period is 24 starting from term 2.
See also
2009 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last question |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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