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Bulgaria National Olympiad
1987 Bulgaria National Olympiad
Problem 4
perfect square recurrence
perfect square recurrence
Source: Bulgaria 1987 P4
June 15, 2021
Sequences
algebra
recurrence relation
Problem Statement
The sequence
(
x
n
)
n
∈
N
(x_n)_{n\in\mathbb N}
(
x
n
)
n
∈
N
is defined by
x
1
=
x
2
=
1
x_1=x_2=1
x
1
=
x
2
=
1
,
x
n
+
2
=
14
x
n
+
1
−
x
n
−
4
x_{n+2}=14x_{n+1}-x_n-4
x
n
+
2
=
14
x
n
+
1
−
x
n
−
4
for each
n
∈
N
n\in\mathbb N
n
∈
N
. Prove that all terms of this sequence are perfect squares.
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