MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN L Problems
12
L 12
L 12
Source:
May 25, 2007
Linear Recurrences
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
=
12
,
a
3
=
20
,
a
n
+
3
=
2
a
n
+
2
+
2
a
n
+
1
−
a
n
.
a_{1}=1, \; a_{2}=12, \; a_{3}=20, \; a_{n+3}= 2a_{n+2}+2a_{n+1}-a_{n}.
a
1
=
1
,
a
2
=
12
,
a
3
=
20
,
a
n
+
3
=
2
a
n
+
2
+
2
a
n
+
1
−
a
n
.
Prove that
1
+
4
a
n
a
n
+
1
1+4a_{n}a_{n+1}
1
+
4
a
n
a
n
+
1
is a square for all
n
∈
N
n \in \mathbb{N}
n
∈
N
.
Back to Problems
View on AoPS