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Putnam
2018 Putnam
B4
B4
Part of
2018 Putnam
Problems
(1)
Putnam 2018 B4
Source:
12/2/2018
Given a real number
a
a
a
, we define a sequence by
x
0
=
1
x_0 = 1
x
0
=
1
,
x
1
=
x
2
=
a
x_1 = x_2 = a
x
1
=
x
2
=
a
, and
x
n
+
1
=
2
x
n
x
n
−
1
−
x
n
−
2
x_{n+1} = 2x_nx_{n-1} - x_{n-2}
x
n
+
1
=
2
x
n
x
n
−
1
−
x
n
−
2
for
n
≥
2
n \ge 2
n
≥
2
. Prove that if
x
n
=
0
x_n = 0
x
n
=
0
for some
n
n
n
, then the sequence is periodic.
Putnam
Putnam 2018