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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==
 +
(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:
 +
 +
 +
<math>a_3 = \frac{1 + \frac{ \sqrt{3} } {3} } {1-\frac{\sqrt{3}}{3}}</math>
 +
 +
 +
<math>\Rightarrow \frac{3+\sqrt{3}}{3-\sqrt{3}}</math>
 +
 +
 +
<math>\Rightarrow 2 + \sqrt{3}</math>
 +
 +
 +
<math>a_4 = \frac{\frac{\sqrt{3}}{3}+2+\sqrt{3}}{1-\frac{\sqrt{3}(2+\sqrt{3})}{3}}</math>
 +
 +
 +
<math>\Rightarrow \frac{6+4\sqrt{3}}{-2\sqrt{3}}</math>
 +
 +
 +
<math>\Rightarrow -(2+\sqrt{3})</math>
 +
 +
 +
So at this point, our sequence reads
 +
 +
 +
<math>1,\: \frac{\sqrt{3}}{3}, \: 2 + \sqrt{3}, \: -(2+\sqrt{3})</math>
 +
 +
 +
Now for <math>a_5</math>..... but wait! the numerator of the next term is equal to <math>2 + \sqrt{3} + -(2+\sqrt{3})</math>..... <math>\Rightarrow 0</math>. So as long as the denominator isn't <math>0</math> (which we can quickly verify), <math>a_5 = 0</math>. Now our sequence is
 +
 +
 +
<math>1,\: \frac{\sqrt{3}}{3}, \: 2 + \sqrt{3}, \: -(2+\sqrt{3}), \: 0</math>
 +
 +
 +
Solving for <math>a_6</math>:
 +
 +
 +
<math>\frac{-(2+\sqrt{3})+0}{1-0}</math>
 +
 +
 +
<math>\Rightarrow -(2+\sqrt{3})</math>
 +
 +
 +
Our sequence is now
 +
 +
 +
<math>1,\: \frac{\sqrt{3}}{3}, \: 2 + \sqrt{3}, \: -(2+\sqrt{3}), \: 0, \: -(2+\sqrt{3})</math>
 +
 +
 +
At this point you should be getting suspicious, because some terms are repeating. Notice that the formula asks for a sum and then later a product of two terms, neither of which depend on order (commutative and associative properties). So that means if <math>a_n = x</math>, and <math>a_{n+1} = y</math>, then we would get the same thing if <math>a_n = y</math> and <math>a_{n+1} = x</math> (Basically, as long the combination of the previous two elements is the same, we should get the same result). So <math>a_7 = 0</math>, <math>a_8 = -(2+\sqrt{3})</math>, <math>a_9 = 0</math>, and so on.
 +
We can generalize this to say
 +
 +
 +
"For all even <math>n</math> such that <math>n \ge 4</math>, <math>a_n = -(2+\sqrt{3})</math>"
 +
 +
"For all odd <math>n</math> such that <math>n \ge 5</math>, <math>a_n = 0</math>"
 +
 +
 +
The problem asks us for the value of <math>a_2009</math>, and since <math>2009</math> is an odd number, we know that <math>a_2009 = 0 \Rightarrow \boxed{\text{E}}</math>.
  
 
== See also ==
 
== See also ==

Revision as of 09:52, 30 July 2020

Problem

The first two terms of a sequence are $a_1 = 1$ and $a_2 = \frac {1}{\sqrt3}$. For $n\ge1$,

\[a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.\]

What is $|a_{2009}|$?

$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 2 - \sqrt3\qquad \textbf{(C)}\ \frac {1}{\sqrt3}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2 + \sqrt3$

Solution

Consider another sequence $\{\theta_1, \theta_2, \theta_3...\}$ such that $a_n = \tan{\theta_n}$, and $0 \leq \theta_n < 180$.

The given recurrence becomes

\begin{align*} a_{n + 2} & = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}} \\ \tan{\theta_{n + 2}} & = \frac {\tan{\theta_n} + \tan{\theta_{n + 1}}}{1 - \tan{\theta_n}\tan{\theta_{n + 1}}} \\ \tan{\theta_{n + 2}} & = \tan(\theta_{n + 1} + \theta_n) \end{align*}

It follows that $\theta_{n + 2} \equiv \theta_{n + 1} + \theta_{n} \pmod{180}$. Since $\theta_1 = 45, \theta_2 = 30$, all terms in the sequence $\{\theta_1, \theta_2, \theta_3...\}$ will be a multiple of $15$.

Now consider another sequence $\{b_1, b_2, b_3...\}$ such that $b_n = \theta_n/15$, and $0 \leq b_n < 12$. The sequence $b_n$ satisfies $b_1 = 3, b_2 = 2, b_{n + 2} \equiv b_{n + 1} + b_n \pmod{12}$.

As the number of possible consecutive two terms is finite, we know that the sequence $b_n$ is periodic. Write out the first few terms of the sequence until it starts to repeat.

$\{b_n\} = \{3,2,5,7,0,7,7,2,9,11,8,7,3,10,1,11,0,11,11,10,9,7,4,11,3,2,5,7,...\}$

Note that $b_{25} = b_1 = 3$ and $b_{26} = b_2 = 2$. Thus $\{b_n\}$ has a period of $24$: $b_{n + 24} = b_n$.

It follows that $b_{2009} = b_{17} = 0$ and $\theta_{2009} = 15 b_{2009} = 0$. Thus $a_{2009} = \tan{\theta_{2009}} = \tan{0} = 0.$

Our answer is $|a_{2009}| = \boxed{\textbf{(A)}\ 0}$.

Solution 2

(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:


$a_3 = \frac{1 + \frac{ \sqrt{3} } {3} } {1-\frac{\sqrt{3}}{3}}$


$\Rightarrow \frac{3+\sqrt{3}}{3-\sqrt{3}}$


$\Rightarrow 2 + \sqrt{3}$


$a_4 = \frac{\frac{\sqrt{3}}{3}+2+\sqrt{3}}{1-\frac{\sqrt{3}(2+\sqrt{3})}{3}}$


$\Rightarrow \frac{6+4\sqrt{3}}{-2\sqrt{3}}$


$\Rightarrow -(2+\sqrt{3})$


So at this point, our sequence reads


$1,\: \frac{\sqrt{3}}{3}, \: 2 + \sqrt{3}, \: -(2+\sqrt{3})$


Now for $a_5$..... but wait! the numerator of the next term is equal to $2 + \sqrt{3} + -(2+\sqrt{3})$..... $\Rightarrow 0$. So as long as the denominator isn't $0$ (which we can quickly verify), $a_5 = 0$. Now our sequence is


$1,\: \frac{\sqrt{3}}{3}, \: 2 + \sqrt{3}, \: -(2+\sqrt{3}), \: 0$


Solving for $a_6$:


$\frac{-(2+\sqrt{3})+0}{1-0}$


$\Rightarrow -(2+\sqrt{3})$


Our sequence is now


$1,\: \frac{\sqrt{3}}{3}, \: 2 + \sqrt{3}, \: -(2+\sqrt{3}), \: 0, \: -(2+\sqrt{3})$


At this point you should be getting suspicious, because some terms are repeating. Notice that the formula asks for a sum and then later a product of two terms, neither of which depend on order (commutative and associative properties). So that means if $a_n = x$, and $a_{n+1} = y$, then we would get the same thing if $a_n = y$ and $a_{n+1} = x$ (Basically, as long the combination of the previous two elements is the same, we should get the same result). So $a_7 = 0$, $a_8 = -(2+\sqrt{3})$, $a_9 = 0$, and so on. We can generalize this to say


"For all even $n$ such that $n \ge 4$, $a_n = -(2+\sqrt{3})$"

"For all odd $n$ such that $n \ge 5$, $a_n = 0$"


The problem asks us for the value of $a_2009$, and since $2009$ is an odd number, we know that $a_2009 = 0 \Rightarrow \boxed{\text{E}}$.

See also

2009 AMC 12A (Problems • Answer Key • Resources)
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