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Fibonacci Sequence so confused

  1. Apr 17, 2004 #1
    My professor really threw a curve ball at me the other day with this problem, and unfortuneately, I have no idea where to begin. Any help would be greatly appreciated.

    In the following problem, all the n's are subdomains of the leading coefficent, along with the f#. f(sub 0), f(sub 1), etc..f(sub n+2), f(sub n+1), f(sub n)...a(sub n)


    Let fn be the Fibonacci sequence, i.e.:
    {f0 = f1 = 1
    {fn+2 = fn+1 + fn, for every n >= 0

    Define now a new sequence, {an}, given by an = fn+1 / fn

    (a) show that an+1 = 1 + (1/an)
    (b) assuming that {an}infinity n=1 is convergent, find lim(n->inf) an.

    Please offer any kind of assistance you can. Thanks
     
  2. jcsd
  3. Apr 18, 2004 #2
    The first one is just basic algebra. You were given a formula for a_n, use it and see if you can "transform" the LHS into something which looks like the RHS.
     
  4. Apr 18, 2004 #3

    matt grime

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    If a_n onverges to a, say, then take limits in the equation in part (a) above to find a.
     
  5. Apr 18, 2004 #4

    marcus

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    noboost4you I will give you a piece of advice that might help a lot (or might not depending on you)

    if you aint no math whiz then whenever possible (time permitting) experiment with real numbers and a calculator

    (dont let them force you to think abstractly and generally before youre ready)

    the Fibs are:
    1,1,2,3,5,8,13,21,.....


    he wants you to study the ratios

    1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13,....

    he wants you to check that

    this sequence approaches a number X with the special
    nice feature that
    X = 1 + 1/X

    multiplying thru by X you see that another way to write that is
    X2 = X + 1

    it wouldnt be true exactly for 21/13, or for any one ratio, but
    it might be almost true for 21/13

    and if you crank out some larger Fibs and take a corresponding ratio furtherout in the sequence it should be closer to being true for that

    If you can figure what number satisfies the equation exactly
    that is what X has
    X2 = X + 1
    exactly
    then you can tell what the successive ratios of Fibs are going to get closer and closer to

    he doesnt want this for an answer, he has a special fancypants way of finding it out
    which you are supposed to step thru, like a trained poodle in the circus.
    but this is the gist of it
    the successive ratios in the Fib sequence go to the Golden Mean
    and you can find this out with a ten buck calculator
     
  6. Apr 18, 2004 #5

    still lost, im sorry. any other pointers?
     
  7. Apr 18, 2004 #6
    sorry, i posted my last message before i saw your response. and to tell you the truth, your response helped me alot. i understand what my professor is now asking. thanks
     
  8. Apr 18, 2004 #7

    marcus

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    heh heh consider that you just got a boost
    now comes the part of figuring out what el proffo
    wants to see on dah homework paper
    good luck
    BTW they are pretty good at helping if you go down
    to college level homework help, near the bottom of
    the index page, or so I thought when I looked in there
    someone named Doc Al, I think
    you can keep asking questions and eventually something may click
     
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