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Math Help - Mathematical Induction Help

  1. #1
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    Mathematical Induction Help

    I have trouble with following question and I was wondering if someone can help me.

    For
    recurrence relation an = 2an-1 +an-2, a1 = 5 and a2 = 10. Use mathematical induction to show that an < 3^nfor all n >3.

    This is how I tried.

    Basic step p(3) = 2(10) + 5 = 25 < 3^3

    So p(3) is true

    Inductive step

    an < 3^n
    2an + ak-1 < 2*3^n + ak-1
    ak+1 < 2*3^n + ak-1

    I would appreciate any help.
    Thanks

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  2. #2
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    a_K  < 3^K \, \Rightarrow \,a_{K + 1}  = 2a_K  + a_{K - 1}  < 2\left( {3^K } \right) + 3^{K - 1}  < 2\left( {3^K } \right) + 3^K  = 3(3^K )
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