2006 AMC 10B Problem 18

Problem 18 of 25IntermediateAlgebra

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}?

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Solution

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.

Concepts: recursion · pattern recognition

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