AMC10 2006 B

AMC10 2006 B · Q18

AMC10 2006 B · Q18. It mainly tests Sequences & recursion (algebra), Sequences in number theory (remainders patterns).

Let $a_1, a_2, \dots$ be a sequence for which $a_1 = 2$, $a_2 = 3$, and $a_n = \frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$?

设序列 $a_1, a_2, \dots$ 满足 $a_1 = 2$，$a_2 = 3$，且对每个正整数 $n \ge 3$，$a_n = \frac{a_{n-1}}{a_{n-2}}$。 $a_{2006}$ 是多少？

(A) 1/2 1/2

(B) 2/3 2/3

(D) 2 2

(E) 3 3

Answer

Correct choice: (E)

正确答案：（E）

Solution

18. (E) Note that the first several terms of the sequence are: $2,\ 3,\ \frac{3}{2},\ \frac{1}{2},\ \frac{1}{3},\ \frac{2}{3},\ 2,\ 3,\ \ldots,$ so the sequence consists of a repeating cycle of 6 terms. Since $2006=334\cdot 6+2$, we have $a_{2006}=a_2=3$.

18.（E）注意到该数列的前几项为： $2,\ 3,\ \frac{3}{2},\ \frac{1}{2},\ \frac{1}{3},\ \frac{2}{3},\ 2,\ 3,\ \ldots,$ 因此该数列由一个长度为 6 的循环重复构成。由于 $2006=334\cdot 6+2$，所以 $a_{2006}=a_2=3$。

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