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National and Regional Contests
Greece Contests
Greece National Olympiad
2018 Greece National Olympiad
1
1
Part of
2018 Greece National Olympiad
Problems
(1)
First term is a square
Source: 2018 Greece National Olympiad Problem 1
3/3/2018
Let
(
x
n
)
,
n
∈
N
(x_n), n\in\mathbb{N}
(
x
n
)
,
n
∈
N
be a sequence such that
x
n
+
1
=
3
x
n
3
+
x
n
,
∀
n
∈
N
x_{n+1}=3x_n^3+x_n, \forall n\in\mathbb{N}
x
n
+
1
=
3
x
n
3
+
x
n
,
∀
n
∈
N
and
x
1
=
a
b
x_1=\frac{a}{b}
x
1
=
b
a
where
a
,
b
a,b
a
,
b
are positive integers such that
3
∤
b
3\not|b
3
∣
b
. If
x
m
x_m
x
m
is a square of a rational number for some positive integer
m
m
m
, prove that
x
1
x_1
x
1
is also a square of a rational number.
Sequence
Perfect Square