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National and Regional Contests
PEN Problems
PEN M Problems
18
18
Part of
PEN M Problems
Problems
(1)
M 18
Source:
5/25/2007
Given is an integer sequence
{
a
n
}
n
≥
0
\{a_n\}_{n \ge 0}
{
a
n
}
n
≥
0
such that
a
0
=
2
a_{0}=2
a
0
=
2
,
a
1
=
3
a_{1}=3
a
1
=
3
and, for all positive integers
n
≥
1
n \ge 1
n
≥
1
,
a
n
+
1
=
2
a
n
−
1
a_{n+1}=2a_{n-1}
a
n
+
1
=
2
a
n
−
1
or
a
n
+
1
=
3
a
n
−
2
a
n
−
1
a_{n+1}= 3a_{n} - 2a_{n-1}
a
n
+
1
=
3
a
n
−
2
a
n
−
1
. Does there exist a positive integer
k
k
k
such that
1600
<
a
k
<
2000
1600 < a_{k} < 2000
1600
<
a
k
<
2000
?
induction
Recursive Sequences