Fibonacci sequences
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
[email protected]
Webpage
http://www.math.ufl.edu/
∼
squash/
26 April, 2009
(at
12:08
)
Prolegomenon.
The famous
Fibonacci sequence
~
f
:= (
f
n
)
∞
n
=
∞
is defined by
f
0
:= 0,
f
1
:= 1 and
f
n
+1
=
f
n
+
f
n

1
.
1
:
Let
P
and
G
be the Positive and neGative roots of
Fib(
x
) :=
x
2

x

1
note
==== [
x

P
][
x

G
]
.
So
P
+
G
= 1
and
P
·
G
=
1
.
Moreover,
P
2
=
P
+ 1
and
G
2
=
G
+ 1
.
2
:
Let
ρ
:=
√
5 and
b
:=
1
√
5
. Certainly
~
f
is some linear
combination
α
·
[
n
7→
P
n
] +
β
·
[
n
7→
G
n
]. Thus
∀
n
∈
Z
:
f
n
=
b
·
[
P
n

G
n
]
,
3
:
since this formula gives correct values for
f
0
and
f
1
.
4
:
Theorem.
For each integer
N
:
[
f
N
]
2
+ [
f
N

1
]
2
=
f
2
N

1
.
4a
:
♦
Proof.
Always, LhS(4
a
) is nonnegative. And RhS(4
a
)
is nonnegative, even when
N
∈
Z

, since
f
Odd
is
always nonnegative. So ISTProve that the squares of
LhS(4
a
) and RhS(4
a
) are equal. To this end, define
R
:=
ρ
2
·
[
f
2
N

1
]
note
====
ρ
·
[
P
2
N

1

G
2
N

1
] and
L
:=
ρ
2
·
[
f
N
]
2
+ [
f
N

1
]
2
.
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 Fall '07
 JURY
 Math, Calculus, Fibonacci number, FN, famous Fibonacci Sequence, Webpage http://www.math.uﬂ.edu/∼squash/, fj fk−1, fk + fj

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