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Fibonacci Sequence

Date: 01/29/2001 at 22:05:04
From: Alisha Winters
Subject: Fibonacci Sequence

I'm taking a class for teaching math to elementary students, and we 
did a problem with the Fibonacci sequence. Afterward our teacher 
challenged each of us to find out if there is a formula to find the 
first "n" terms of the sequence. Math is not my strongest subject so 
I've come to you to see if you can help. Thanks!

Date: 01/29/2001 at 22:30:30
From: Doctor Rob
Subject: Re: Fibonacci Sequence

Thanks for writing to Ask Dr. Math, Alisha.

Yes, there is a formula for the n-th Fibonacci number.  It is:

   F(n) = [([1+sqrt(5)]/2)^n - ([1-sqrt(5)]/2)^n]/sqrt[5].

Now it may seem impossible that this complicated formula involving
sqrt(5) would evaluate to an integer, but it does.

   F(0) = [([1+sqrt(5)]/2)^0 - ([1-sqrt(5)]/2)^0]/sqrt[5],
        = [1 - 1]/sqrt[5] = 0.

   F(1) = [([1+sqrt(5)]/2)^1 - ([1-sqrt(5)]/2)^1]/sqrt[5],
        = [1/2+sqrt(5)/2-1/2+sqrt(5)/2]/sqrt[5],
        = sqrt[5]/sqrt[5] = 1.

   F(2) = [([1+sqrt(5)]/2)^2 - ([1-sqrt(5)]/2)^2]/sqrt[5],
        = [(3+sqrt[5])/2 - (3-sqrt[5])/2]/sqrt[5],
        = sqrt[5]/sqrt[5] = 1.

You can see that the square roots always cancel out, which seems a
bit miraculous, but true!

- Doctor Rob, The Math Forum   
Associated Topics:
High School Fibonacci Sequence/Golden Ratio
High School Number Theory

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