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Fibonacci limit

  1. Jan 24, 2007 #1
    Let:
    [tex]a_{1}=a_{2}=1;a_{n+2}=a_{n+1}+a_{n};n\geq 1 [/tex]

    Let [itex]f_{n}[/itex] be the last digit in decimal notation
    of Fibonacci number [itex]a_{n}[/itex].
    Find:

    [tex]\lim_{n\to\infty}\frac{a_{1}+a_{2}+...+a_{n}}{n}[/tex]
     
    Last edited: Jan 24, 2007
  2. jcsd
  3. Jan 24, 2007 #2
    Can you explain?
     
  4. Jan 24, 2007 #3

    mathman

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    What does fn have to do with anything???
     
  5. Jan 24, 2007 #4
    By theory, there is no limit to Fibonacci, unless I'm mistaken.

    The sequence wouldn't be a sequence if there was a limit.
     
  6. Jan 24, 2007 #5

    HallsofIvy

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    That doesn't quite make sense. A fair part of Calculus courses is devoted to limits of sequences! Of course, the Fibonacci sequence is increasing without upperbound so it has no limit. But the question is about the nth partial sum divided by n. That's a whole different matter.
     
  7. Jan 24, 2007 #6

    matt grime

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    What the hell is anyone talking about here?
     
  8. Jan 24, 2007 #7
    That's what I was getting at, thanks for clarifying!
     
  9. Jan 24, 2007 #8

    CRGreathouse

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    The question is about the final digits, which are periodic with period 60. The sum of the 60 values is ***, so the average value at the limit is ***/60.

    (It's not hard to calculate this, so I left it as an exercise. I can check it if you think you have an answer.)
     
  10. Jan 25, 2007 #9
    correction (+ solution)

    Let:
    [tex]a_{1}=a_{2}=1;a_{n+2}=a_{n+1}+a_{n};n\geq 1 [/tex]

    Let [itex]f_{n}[/itex] be the last digit in decimal notation
    of Fibonacci number [itex]a_{n}[/itex].

    Find:

    [tex]\lim_{n\to\infty}\frac{f_{1}+f_{2}+...+f_{n}}{n}[/tex]

    My apology for the confusion I made.


    EDIT:
    Yes the key for the solution is "***/60".
    IOW ,[itex]f_{1}=f_{61},f_{2}=..etc.[/itex]
    I get:
    [tex]\lim_{n\to\infty}\frac{f_{1}+f_{2}+...+f_{n}}{n}=\frac{14}{3}[/tex]
     
    Last edited: Jan 25, 2007
  11. Jan 25, 2007 #10

    CRGreathouse

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    Yes, 14/3 is right.
     
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