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Math Help - induction on a reccurence sequence

  1. #1
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    induction on a reccurence sequence

    i have a1=1

    a(n+1) = (5an+4)/(an+2) for n>=1

    need to prove by induction on n that 0<an<4 .

    course of action would be?

    then n is subscript and (n+1) also subscript , i think i need to get it into the form an+2 =A an+1 + B an

    where A and B are some fixed numbers?

    thanks
    Last edited by ad53ggz; November 29th 2008 at 07:39 AM.
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  2. #2
    MHF Contributor
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    Hello !

    The start of the recurrence is OK since 0 < a1=1 < 4

    Suppose that 0 < a_n < 4
    Then
    a_{n+1}\;=\;\frac{5a_n+4}{a_n+2}

    a_{n+1}\;=\;5-\frac{6}{a_n+2}

    0 < a_n < 4

    2 < a_n +2 < 6

    \frac{1}{6} < \frac{1}{a_n+2} < \frac{1}{2}

    1 < \frac{6}{a_n+2} < 3

    2 < 5-\frac{6}{a_n+2} < 4

    0 < 2 < a_{n+1} < 4
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  3. #3
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    an+1 = 5-6/an+2





    spot on mate was this bit i couldnt work out feel dumb now
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