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National and Regional Contests
PEN Problems
PEN L Problems
5
L 5
L 5
Source:
May 25, 2007
Linear Recurrences
Problem Statement
The Fibonacci sequence
{
F
n
}
\{F_{n}\}
{
F
n
}
is defined by
F
1
=
1
,
F
2
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
.
F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.
F
1
=
1
,
F
2
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
.
Show that
F
2
n
−
1
2
+
F
2
n
+
1
2
+
1
=
3
F
2
n
−
1
F
2
n
+
1
F_{2n-1}^{2}+F_{2n+1}^{2}+1=3F_{2n-1}F_{2n+1}
F
2
n
−
1
2
+
F
2
n
+
1
2
+
1
=
3
F
2
n
−
1
F
2
n
+
1
for all
n
≥
1
n \ge 1
n
≥
1
.
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