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Fibonacci Mystery


A new friend was recently showing me an interesting mystery using the Fibonacci Numbers that I am at a loss to explain. I was hoping to post it here to see if someone else can help solve this mystery with a proof or explanation.

First, let's define n as an integer number greater than zero. And F(n) as the nth number of the Fibonacci series. In other words:


nF(n)
1
1
2
1
3
2
4
3
5
5
6
8
7
13
8
21
9
34
10
55
Table #1


As you can see from above, the definition of a Fibonacci number is the sum of the two previous Fibonacci. This can be written as:

F(n+2) = F(n+1) + F(n)

This can be better shown as:


nF(n)F(n+1)F(n+2)
1
1
1
2
2
1
2
3
3
2
3
5
4
3
5
8
5
5
8
13
6
8
13
21
7
13
21
34
8
21
34
55
9
34
55
89
10
55
89
144
Table #2


Now here is where an interesting pattern starts appearing that I hope someone can explain to me. Assume x is the multiplication of two Fibonacci numbers that only has a single Fibonacci number between them. This can be written as x = F(n) * F(n+2).




And assume y is the the middle Fibonacci number between the previous two Fibonacci numbers times itself. This can be written as y = F(n+1) ^ 2. Finally, z is the difference between x and y.

This can be better shown as:


nF(n)F(n+1)F(n+2)x = F(n) * F(n+2)y = F(n+1) ^ 2z = x - y
1
1
1
2
2
1
1
2
1
2
3
3
4
-1
3
2
3
5
10
9
1
4
3
5
8
24
25
-1
5
5
8
13
65
64
1
6
8
13
21
168
169
-1
7
13
21
34
442
441
1
8
21
34
55
1155
1156
-1
9
34
55
89
3026
3025
1
10
55
89
144
7920
7921
-1
Table #3


Notice that z alternates between -1 and 1.

Also, this is true:


nz^2 = 1z = -1^(n+1)
1
1
1
2
1
-1
3
1
1
4
1
-1
5
1
1
6
1
-1
7
1
1
8
1
-1
19
1
1
10
1
-1
Table #4


While this pattern is very interesting, it is not a proof. Can someone explain or prove to me why this is true?

Someone just emailed me this proof. Let me know if you find anything wrong with it, because I cannot.

Someone else, who also wishes to remain anonymous, sent in another proof using a totally different approach to proving the same thing. Again, if you find anything wrong with it, please let me know.


I already tried proving these equations; but so far, I've had no success.

z = -1^(n+1)

or

(F(n)*F(n+2)) (F(n+1))^2 = -1^(n+1)

I've even tried proving:

z^2 = 1

or

( (F(n)*F(n+2)) (F(n+1))^2 ) ^2 = 1

Thank you very much!

by Phil for Humanity
on 10/01/2011

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