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PAMO Problem 1: Linear recurrence relation
Source: 2019 Pan-African Mathematics Olympiad, Problem 1
4/9/2019
Let
(
a
n
)
n
=
0
∞
(a_n)_{n=0}^{\infty}
(
a
n
)
n
=
0
∞
be a sequence of real numbers defined as follows:[*]
a
0
=
3
a_0 = 3
a
0
=
3
,
a
1
=
2
a_1 = 2
a
1
=
2
, and
a
2
=
12
a_2 = 12
a
2
=
12
; and [*]
2
a
n
+
3
−
a
n
+
2
−
8
a
n
+
1
+
4
a
n
=
0
2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0
2
a
n
+
3
−
a
n
+
2
−
8
a
n
+
1
+
4
a
n
=
0
for
n
≥
0
n \geq 0
n
≥
0
.Show that
a
n
a_n
a
n
is always a strictly positive integer.
recurrence relation
Linear Recurrences
algebra
induction
PAMO