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Math Help - General rule for a sequence..

  1. #1
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    General rule for a sequence..

    hi,

    I am having trouble trying to find a general rule for the sequence defined as a_1 =10, a_{k+1} = \frac{(a_k)^2}{a_k+1}
    I have the first n terms a_1 = 10, a_2 = 4/3, a_3 = 9/4, a_4 = 16/5 .. but cant see a pattern..

    Thanks,
    Last edited by CaptainBlack; October 9th 2011 at 12:25 AM. Reason: Hopefull to fix the LaTeX
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  2. #2
    Grand Panjandrum
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    Re: General rule for a sequence..

    Quote Originally Posted by Oiler View Post
    hi,

    I am having trouble trying to find a general rule for the sequence defined as a_1 =10, a_{k+1} = \frac{(a_k)^2}{a_k+1}
    I have the first n terms a_1 = 10, a_2 = 4/3, a_3 = 9/4, a_4 = 16/5 .. but cant see a pattern..

    Thanks,
    Assuming the correction I made to your LaTeX is correct those are not the terms of the sequence, but those of (other than the first which you were given):

    b_{k}=\frac{k^2}{k+1}

    Assuming the correction:

    a_2=\frac{a_1^2}{a_1+1}=\frac{100}{11}

    CB
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  3. #3
    MHF Contributor chisigma's Avatar
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    Re: General rule for a sequence..

    Quote Originally Posted by Oiler View Post
    hi,

    I am having trouble trying to find a general rule for the sequence defined as a_1 =10, a_{k+1} = \frac{(a_k)^2}{a_k+1}
    I have the first n terms a_1 = 10, a_2 = 4/3, a_3 = 9/4, a_4 = 16/5 .. but cant see a pattern..

    Thanks,
    The difference equation...

    a_{n+1}= \frac{a_{n}^{2}}{1+a_{n}}\ ,\ a_{0}=10 (1)

    ... is nonlinear and a direct solution is a difficult task. However, if You are interested to the 'asyntotic behaviour' of the a_{n} then You can write (1) as...

    \Delta_{n}= a_{n+1}-a_{n}= -\frac{a_{n}}{1+a_{n}}= f(a_{n})\ ,\ a_{0}=10 (2)

    ... and follow the procedure illustrated in...

    http://www.mathhelpforum.com/math-he...-i-188482.html

    In Your case is...

    f(x)= -\frac{x}{1-x} (3)

    ... and f(x) has only one 'attractive fixed point' in x_{0}=0. You can verify that for any a_{0} \ne 0 is...

    |f(x)|<|x_{0}-x| (4)

    ... so that the conditions of theorem 4.1 are satisfied and the a_{n} do converge monotonically at 0...

    Kind regards

    \chi \sigma
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