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2007 AIME Problems
14
Recursive Sequence
Recursive Sequence
Source: AIME I 2007 #14
March 15, 2007
AMC
Problem Statement
Let a sequence be defined as follows:
a
1
=
3
a_{1}= 3
a
1
=
3
,
a
2
=
3
a_{2}= 3
a
2
=
3
, and for
n
≥
2
n \ge 2
n
≥
2
,
a
n
+
1
a
n
−
1
=
a
n
2
+
2007
a_{n+1}a_{n-1}= a_{n}^{2}+2007
a
n
+
1
a
n
−
1
=
a
n
2
+
2007
. Find the largest integer less than or equal to
a
2007
2
+
a
2006
2
a
2007
a
2006
\frac{a_{2007}^{2}+a_{2006}^{2}}{a_{2007}a_{2006}}
a
2007
a
2006
a
2007
2
+
a
2006
2
.
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