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Math Help - inequality induction question

  1. #1
    Junior Member
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    inequality induction question

    Having some trouble with this one...

    Consider the sequence a(n) n in natural numbers defined by a(1) = 1, a(2) = 2, a(3) = 3 and a(n) = a(n-1) + a(n-2)+a(n-3) for all n in natural numbers satisfying n>3. Use induction on n to show that a(n) <= 3^n for and n in natural numbers.

    Now I can get up to inductive hypothesis but I am stuck after that.

    Suppose a(k) <= 3^k is true...

    therefore a(k+1) <= 3^(k+1) must be true too.
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  2. #2
    Member Traveller's Avatar
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    a(k+1) = a(k) + a(k-1) + a(k-2) <= 3^k + 3^(k-1) + 3^(k-2) < 3^k +3^k +3^k = 3* (3^k) = 3^(k+1)
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