Difference between revisions of "2009 AMC 12A Problems/Problem 25"
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Our answer is <math>|a_{2009}| = \boxed{\textbf{(A)}\ 0}</math>. | Our answer is <math>|a_{2009}| = \boxed{\textbf{(A)}\ 0}</math>. | ||
− | + | == Solution 2 == | |
First note how this sequence can be rewritten as tans, and each term is tan of sum of previous 2 angles. Thus we can ignore the tan for now and just focus on the angle itself; first few are 45, 30, 75, 105, 180, and since the question is asking for absolute value of the 2009th term, 180°=0°. This means after the first 180° each successive odd term mod 180 is 0 and each even term is 105 mod 180. Since 2009 is odd, you find tan(0) which is just 0. | First note how this sequence can be rewritten as tans, and each term is tan of sum of previous 2 angles. Thus we can ignore the tan for now and just focus on the angle itself; first few are 45, 30, 75, 105, 180, and since the question is asking for absolute value of the 2009th term, 180°=0°. This means after the first 180° each successive odd term mod 180 is 0 and each even term is 105 mod 180. Since 2009 is odd, you find tan(0) which is just 0. | ||
Revision as of 12:52, 11 January 2017
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
First note how this sequence can be rewritten as tans, and each term is tan of sum of previous 2 angles. Thus we can ignore the tan for now and just focus on the angle itself; first few are 45, 30, 75, 105, 180, and since the question is asking for absolute value of the 2009th term, 180°=0°. This means after the first 180° each successive odd term mod 180 is 0 and each even term is 105 mod 180. Since 2009 is odd, you find tan(0) which is just 0.
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) | |
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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