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National and Regional Contests
The Philippines Contests
Philippine MO
2020 Philippine MO
3
3
Part of
2020 Philippine MO
Problems
(1)
Raving recurrence relation
Source: Philippine Mathematical Olympiad 2020/3
1/19/2020
Define the sequence
{
a
i
}
\{a_i\}
{
a
i
}
by
a
0
=
1
a_0=1
a
0
=
1
,
a
1
=
4
a_1=4
a
1
=
4
, and
a
n
+
1
=
5
a
n
−
a
n
−
1
a_{n+1}=5a_n-a_{n-1}
a
n
+
1
=
5
a
n
−
a
n
−
1
for all
n
≥
1
n\geq 1
n
≥
1
. Show that all terms of the sequence are of the form
c
2
+
3
d
2
c^2+3d^2
c
2
+
3
d
2
for some integers
c
c
c
and
d
d
d
.
PMO
number theory