Question

Los primeros dos términos de una sucesión son $a_1 = 1$ y $a_2 = \dfrac{1}{\sqrt{3}}.$ Para $n \ge 1,$ $a_{n+2} = \dfrac{a_n + a_{n+1}}{1 - a_n a_{n+1}}.$ ¿Cuánto vale $|a_{2009}|$ ?

The first two terms of a sequence are $a_1 = 1$ and $a_2 = \dfrac{1}{\sqrt{3}}.$ For $n \ge 1,$ $a_{n+2} = \dfrac{a_n + a_{n+1}}{1 - a_n a_{n+1}}.$ What is $|a_{2009}|?$

$0$

$2 - \sqrt{3}$

$\dfrac{1}{\sqrt{3}}$

$1$

$2 + \sqrt{3}$

Solución:

La recursión es exactamente la fórmula de la tangente de una suma, y $a_1 = \tan\dfrac{\pi}{4},$ $a_2 = \tan\dfrac{\pi}{6}.$

Escribiendo $a_n = \tan\dfrac{\pi c_n}{12}$ con $c_1 = 3,$ $c_2 = 2,$ y $c_{n+2} \equiv c_n + c_{n+1} \pmod{12},$ la sucesión $c_n$ es $\begin{gathered} 3, 2, 5, 7, 0, 7, 7, 2, \\ 9, 11, 8, 7, 3, 10, 1, 11, \\ 0, 11, 11, 10, 9, 7, 4, 11, \ldots \end{gathered}$ que es periódica con período $24.$

Como $2009 = 24\cdot 83 + 17,$ $c_{2009} = c_{17} = 0,$ así que $a_{2009} = \tan 0 = 0$ y $|a_{2009}| = 0.$

Por lo tanto, la respuesta correcta es A.

The recursion is exactly the tangent addition formula, and $a_1 = \tan\dfrac{\pi}{4},$ $a_2 = \tan\dfrac{\pi}{6}.$

Writing $a_n = \tan\dfrac{\pi c_n}{12}$ with $c_1 = 3,$ $c_2 = 2,$ and $c_{n+2} \equiv c_n + c_{n+1} \pmod{12},$ the sequence $c_n$ is $\begin{gathered} 3, 2, 5, 7, 0, 7, 7, 2, \\ 9, 11, 8, 7, 3, 10, 1, 11, \\ 0, 11, 11, 10, 9, 7, 4, 11, \ldots \end{gathered}$ which is periodic with period $24.$

Since $2009 = 24\cdot 83 + 17,$ $c_{2009} = c_{17} = 0,$ so $a_{2009} = \tan 0 = 0$ and $|a_{2009}| = 0.$

Thus, the correct answer is A.

2009 AMC 12A Problema 25

Solución:

El Problema 25 en otros años