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Fibonacci Numbers: Understand one part not another

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I understand how to use a calculator to write the decimal point using the golden ratio which is 1.618034 and the conjugate is -.618034. I also know that the relationship between the decimal expansions is that everything to the right of the decimal is the same except one is neg. and one is pos.

Now I have a problem. I have to use the Binet form to calculate the 20th Fib. number. I can use my decimals that I used in the problem above, but I am not sure how to do this. I have worked out the Fibonacci numbers and found that the 20th number is 6,765.
I could really use some help in understanding my problem.

  • Fibonacci Numbers: Understand one part not another -

    You have correctly calculated the 20th fibonacci number as 6765. That's an excellent start.

    To calculate any Fibonacci number without having to go through the recursive process (n times) is to use the Binet's formula.

    To simiply things, two constants have been defined:
    φ=(1+sqrt(5))/2, and
    ψ=(1-sqrt(5))/2
    then the nth Fibonacci number can be found with the following formula:
    F(n)=(φ^n-ψ^n)/sqrt(5)

    If your calculator does not have constants to store φ and ψ you can calculate directly as:

    F(n)=(((1+√5)/2)^n-((1-√5)/2)^n)/√5

    It should give 6765 if you put n=20.

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