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Thread: Mathematical Induction :)

  1. June 26th 2012, 05:09 AM #1
    Doll0555
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    Question Mathematical Induction :)

    solve using mathematical induction: 2 times 34^n -3 times 23^n +1 is divisible by 726 for all positive integers n.
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  2. June 26th 2012, 06:16 AM #2
    emakarov
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    Re: Mathematical Induction :)

    If 2\cdot34^n - 3\cdot23^n + 1 = 726k, then 68\cdot34^n-102\cdot23^n+34=726k', i.e., 68\cdot34^n = 102\cdot23^n-34+726k'. Therefore,

    2\cdot34^{n+1} - 3\cdot23^{n+1} + 1 = 68\cdot34^n - 69\cdot23^n + 1 =
    33\cdot23^n-33+726k' = 33(23^n-1) + 726k'.

    So, it is sufficient to prove that 23^n-1 is divisible by 22.
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