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Math Help - Increasing sequence

  1. #1
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    Increasing sequence

    Is it possible to show that the sequence defined by
    a_1 = 1, \hskip 2em a_{n+1} = 3-\frac{1}{a_n}
    is increasing without relying on mathematical induction?
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  2. #2
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    Quote Originally Posted by Euclid's Bridge Habitant View Post
    Is it possible to show that the sequence defined by
    a_1 = 1, \hskip 2em a_{n+1} = 3-\frac{1}{a_n}
    is increasing without relying on mathematical induction?
    Almost certainly not, because the result depends on knowing that a_1=1. If for example a_1=4 then the sequence will be decreasing. In order to use information about a_1 to prove something about a_{n+1}-a_n, you need to bridge the gap between 1 and n. The only apparent way to do this is to use an inductive method.
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  3. #3
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    Quote Originally Posted by Opalg View Post
    Almost certainly not, because the result depends on knowing that a_1=1. If for example a_1=4 then the sequence will be decreasing. In order to use information about a_1 to prove something about a_{n+1}-a_n, you need to bridge the gap between 1 and n. The only apparent way to do this is to use an inductive method.
    I might add that if you could solve the difference equation eactly, you might be able to deduce that it is increasing but this is a nonlinear difference equation and they tend to be very difficult to solve exactly.
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