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Cono Sur Olympiad
2003 Cono Sur Olympiad
2
Multiplicative recursion
Multiplicative recursion
Source: Cono Sur 2003 #2
November 18, 2015
number theory
cono sur
Problem Statement
Define the sequence
{
a
n
}
\{a_n\}
{
a
n
}
in the following manner:
a
1
=
1
a_1=1
a
1
=
1
a
2
=
3
a_2=3
a
2
=
3
a
n
+
2
=
2
a
n
+
1
a
n
+
1
a_{n+2}=2a_{n+1}a_{n}+1
a
n
+
2
=
2
a
n
+
1
a
n
+
1
; for all
n
≥
1
n\geq1
n
≥
1
Prove that the largest power of
2
2
2
that divides
a
4006
−
a
4005
a_{4006}-a_{4005}
a
4006
−
a
4005
is
2
2003
.
2^{2003}.
2
2003
.
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