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International Contests
Austrian-Polish
1981 Austrian-Polish Competition
6
6
Part of
1981 Austrian-Polish Competition
Problems
(1)
x_{n+1}=y_n +1/x_n, y_{n+1}=z_n +1/y_n, z_{n+1}=x_n +1/z_n
Source: Austrian Polish 1981 APMC
4/30/2020
The sequences
(
x
n
)
,
(
y
n
)
,
(
z
n
)
(x_n), (y_n), (z_n)
(
x
n
)
,
(
y
n
)
,
(
z
n
)
are given by
x
n
+
1
=
y
n
+
1
x
n
x_{n+1}=y_n +\frac{1}{x_n}
x
n
+
1
=
y
n
+
x
n
1
,
y
n
+
1
=
z
n
+
1
y
n
y_{n+1}=z_n +\frac{1}{y_n}
y
n
+
1
=
z
n
+
y
n
1
,
z
n
+
1
=
x
n
+
1
z
n
z_{n+1}=x_n +\frac{1}{z_n}
z
n
+
1
=
x
n
+
z
n
1
for
n
≥
0
n \ge 0
n
≥
0
where
x
0
,
y
0
,
z
0
x_0,y_0, z_0
x
0
,
y
0
,
z
0
are given positive numbers. Prove that these sequences are unbounded.
recurrence relation
Sequence
unbounded
algebra
bounded