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Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2018 Moldova Team Selection Test
2
2
Part of
2018 Moldova Team Selection Test
Problems
(1)
Interesting problem
Source: Moldova TST 2018
3/5/2018
The sequence
(
a
n
)
n
∈
N
\left(a_{n}\right)_{n\in\mathbb{N}}
(
a
n
)
n
∈
N
is defined recursively as
a
0
=
a
1
=
1
a_{0}=a_{1}=1
a
0
=
a
1
=
1
,
a
n
+
2
=
5
a
n
+
1
−
a
n
−
1
a_{n+2}=5a_{n+1}-a_{n}-1
a
n
+
2
=
5
a
n
+
1
−
a
n
−
1
,
∀
n
∈
N
\forall n\in\mathbb{N}
∀
n
∈
N
Prove that
a
n
∣
a
n
+
1
2
+
a
n
+
1
+
1
a_{n}\mid a_{n+1}^{2}+a_{n+1}+1
a
n
∣
a
n
+
1
2
+
a
n
+
1
+
1
for any
n
∈
N
n\in\mathbb{N}
n
∈
N
number theory