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Fibonacci Numbers

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Laura

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Apr 6, 2001, 9:56:15 AM4/6/01
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Hello. One of the properties of Fibonacci numbers is as stated:

After the first four numbers in the sequence, examination reveals the
following relationship:

(1) The sum of any two adjacent numbers gives the next higher number
in the sequence.

I would like to prove this but am having trouble. Can somebody help?

Here's what I've done so far:

Let F(n) = F(n+1) - F(n-1) for n>=2 where F(1)=F(2)=1

Inductive Step:
Rewrite the above sequence as F(n+1)=F(n)+F(n-1) since this is what we
want to prove. Let n=3
F(4)=3=F(3)+F(2)=2+1=3

Basis Step:
Show that this holds for F(n+1).
Prove that F(n+2)=F(n+1)+F(n)

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

Now what??

Macavity

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Apr 6, 2001, 10:44:16 AM4/6/01
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"Laura" <le...@hotmail.com> wrote in message
news:m1aqks...@forum.mathforum.com...

What is your definition of Fibonacci numbers?


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G.E. Ivey

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Apr 7, 2001, 11:08:29 PM4/7/01
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>Now what??

What definition of Fibonacci numbers are you using? The definition
I've always used is "F(n)= F(n-1)+ F(n-2) if n>2" which is what you
want to prove. You appear to be using F(n)= F(n+1)- F(n-1) as the
definition. If that is the case, you don't need to use induction.
You can immediately get F(n+1)= F(n)+ F(n-1) for all n> 1 which is,
again, what you wanted to prove. If you wish, you can let i= n-1 to
get F(i)= F(i-1)+ F(i-2) which is what I wrote.

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