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2016 AMC 12B Problems/Problem 25

Revision as of 11:00, 21 February 2016 by Zhuangzhuang (talk | contribs) (Created page with "==Problem== The sequence <math>(a_n)</math> is defined recursively by <math>a_0=1</math>, <math>a_1=\sqrt[19]{2}</math>, and <math>a_n=a_{n-1}a_{n-2}^2</math> for <math>n\geq ...")
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Problem

The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer?

$\textbf{(A)}\ 17\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 21$

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