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Math Help - prove sequence is decreasing using PMI

  1. #1
    Junior Member mremwo's Avatar
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    prove sequence is decreasing using PMI

    *x_n indicates x sub n
    *>= indicates greater than or equal to

    Let x_1 >= 2 and x_(n+1) = 1+ sqrt(x_n - 1)

    I can prove that x_n>=x_n+1 --> x_n+1 >= x_n+2 but when I try to do the base case (or what I think is the base case), I can't show x_1>=x>2:

    Please help me show that x_1 >= x_2 for my base case.

    Unless I am doing this completely wrong. This is my first induction problem using sequences, so I have no clue. Thanks!
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  2. #2
    MHF Contributor chisigma's Avatar
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    A similar problem [posted by the same user...] has been solved recently in...

    http://www.mathhelpforum.com/math-he...it-170087.html

    The difference equation can be written as...

    \displaystyle \Delta_{n}= x_{n+1}-x_{n} = 1+\sqrt{x_{n}-1} - x_{n} = f(x_{n}) (1)

    Following the trace of the post I have indicated You should be able to...

    a) find the 'attractive fixed point' x_{0}

    b) find the range of values of x_{1} for which the sequence tends to x_{0}

    c) extablish that the convergence is monotonic or not

    Kind regards

    \chi \sigma
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