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National and Regional Contests
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PEN M Problems
4
M 4
M 4
Source:
May 25, 2007
Euler
Recursive Sequences
Problem Statement
The sequence
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
is defined by
a
1
=
1
,
a
2
=
2
,
a
3
=
24
,
a
n
=
6
a
n
−
1
2
a
n
−
3
−
8
a
n
−
1
a
n
−
2
2
a
n
−
2
a
n
−
3
(
n
≥
4
)
.
a_{1}=1, \; a_{2}=2, \; a_{3}=24, \; a_{n}=\frac{ 6a_{n-1}^{2}a_{n-3}-8a_{n-1}a_{n-2}^{2}}{a_{n-2}a_{n-3}}\ \ \ \ (n\ge4).
a
1
=
1
,
a
2
=
2
,
a
3
=
24
,
a
n
=
a
n
−
2
a
n
−
3
6
a
n
−
1
2
a
n
−
3
−
8
a
n
−
1
a
n
−
2
2
(
n
≥
4
)
.
Show that
a
n
a_{n}
a
n
is an integer for all
n
n
n
, and show that
n
∣
a
n
n|a_{n}
n
∣
a
n
for every
n
∈
N
n\in\mathbb{N}
n
∈
N
.
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