Let a_1, a_2, … be a sequence for which a_1=2, a_2=3, and a_n=a_n-1a_n-2 for each positive integer n≥ 3. What is a_2006?

2006 AMC 10B Problem 18

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18.

Let

a_1,

a_2,

\ldots

be a sequence for which

a_1=2,

a_2=3,

and

a_n=\dfrac{a_{n-1}}{a_{n-2}}

for each positive integer

n\ge 3.

What is

a_{2006}?

\dfrac{1}{2}

\dfrac{2}{3}

\dfrac{3}{2}

2

3

Compute the first several terms to detect a repeating cycle

The sequence repeats every

6

terms; reduce

2006

modulo

6

The terms are

2,\,3,\,\tfrac32,\,\tfrac12,\,\tfrac13,\,\tfrac23,

then

2,\,3,\ldots,

a cycle of length

6.

Since

2006=6\cdot334+2,

we have

a_{2006}=a_2=3.

Thus, the correct answer is E.