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Math Help - Induction

  1. #1
    Junior Member
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    Induction

    Show by induction:

    x_{n+2}=\frac{x_{n}+x_{n+1}}{2}

    x_{1}=0, x_{2}=1,

    that 0\leq x_{n}\leq 1  , n= 1,2,3,...





    If i suppose that 0\leq x_{k+2}=\frac{x_{k}+x_{k+1}}{2}\leq 1 is valid,
    then i want to show that 0\leq x_{k+3}=\frac{x_{k+1}+x_{k+2}}{2}\leq1 is valid by using my assumption. But i can't find a way to do this.
    Maybe this is totally wrong way to do it.


    /regards
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  2. #2
    Super Member girdav's Avatar
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    Re: Induction

    Don't forget to show the result for n=1 and n=2.
    Since 0\leq x_{k+1}\leq 1 and 0\leq x_{k+2}\leq 1 then 0\leq x_{k+1}+x_{k+2}\leq 2. Intuitively, the result is quite clear: if we take two points between 0 and 1, their middle will be between 0 and 1.
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