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Ireland National Math Olympiad
2017 Irish Math Olympiad
5
a_{n+2} = 2a_{n+1} + 41a_n
a_{n+2} = 2a_{n+1} + 41a_n
Source: Irish MO 2017 paper 1 problem 5
December 12, 2022
number theory
recurrence relation
algebra
Problem Statement
The sequence
a
=
(
a
0
,
a
1
,
a
2
,
.
.
.
)
a = (a_0, a_1,a_2,...)
a
=
(
a
0
,
a
1
,
a
2
,
...
)
is defined by
a
0
=
0
,
a
1
=
2
a_0 = 0, a_1 =2
a
0
=
0
,
a
1
=
2
and
a
n
+
2
=
2
a
n
+
1
+
41
a
n
a_{n+2} = 2a_{n+1} + 41a_n
a
n
+
2
=
2
a
n
+
1
+
41
a
n
Prove that
a
2016
a_{2016}
a
2016
is divisible by
2017.
2017.
2017.
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