2009 AMC 12A Problems/Problem 25
Problem
The first two terms of a sequence are and . For ,
What is ?
Solution
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
(This is for the brute force users; kudos to the very intuitive solution above)
First we interpret the formula:
"The next term in the sequence is equivalent to the sum of the previous two divided by 1 minus their product"
Next, we work out the first few terms of the sequence:
So at this point, our sequence reads
Now for ..... but wait! the numerator of the next term is equal to ..... . So as long as the denominator isn't (which we can quickly verify), . Now our sequence is
Solving for :
Our sequence is now
Solving in this way until $a_$ (Error compiling LaTeX. Unknown error_msg)
The problem asks us for the value of , and since is an odd number, we know that .
Clarification of Solution 1
(While I did say the first solution is intuitive, it's kinda hard for people with just a basic knowledge of math, and no knowledge of proofing symbols to understand.)
Notice that the formula looks incredibly similar to the Tangent Addition Formula
Since , let be . Similarly, let be . Then the formula reads
But from the Tangent Addition Formula we know that this is the formula for or , meaning . So, the sequence is simply the sum of the two angle measures. We continue to sum angle measures, like so:
... wait a minute! !
So now we have
.... Ok this is getting tiring. Let's stop and think about whether we can simplify this. Notice that all of the angle measures are a multiple of . Let's express the angle measures as multiples of .
Let .
(Basically, is the angle measure of the corresponding , divided by )
Now we have
But wait... we're dealing with the function, which has a period (recurrence rate) of or . Since we divided the angle measures by , the period is now (which aligns with what we got earlier: ). This means that we can reduce the terms of the sequence based on the function, which returns the remainder after dividing by a certain amount. So now we can say . Now we continue our sequence:
(It's starting to repeat now)
So , and so on. The sequence repeats every 24 terms
See also
2009 AMC 12A (Problems • Answer Key • Resources) | |
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