Problem:
Let a1​,a2​,… be a sequence for which
a1​=2,a2​=3, and an​=an−2​an−1​​ for each positive integer n≥3
What is a2006​?
Answer Choices:
A. 21​
B. 32​
C. 23​
D. 2
E. 3
Solution:
Note that the first several terms of the sequence are:
2,3,23​,21​,31​,32​,2,3,…
so the sequence consists of a repeating cycle of 6 terms. Since 2006=334⋅6+2, we have a2006​=a2​=(E)3​.
The problems on this page are the property of the MAA's American Mathematics Competitions