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Math Help - My other induction question (sequences)

  1. #1
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    My other induction question (sequences)

    I am trying to figure this out.
    I have used . to denote subscript.

    A sequence h.0,h.1,h.2,... is defined as h.0=1, h.1=2, h.2=3
    h.k= h.k-1 + h.k-2 + h.k-3

    show h.n <= 3^n

    I have it to the point where h.k+1 = h.k + h.k-1 + h.k-2
    = 3^k + h.k-1 + h.k-2

    but once again am stuck as to what to do.
    Any help muchly appreciated
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by R.Laramee View Post
    I am trying to figure this out.
    I have used . to denote subscript.

    A sequence h.0,h.1,h.2,... is defined as h.0=1, h.1=2, h.2=3
    h.k= h.k-1 + h.k-2 + h.k-3

    show h.n <= 3^n

    I have it to the point where h.k+1 = h.k + h.k-1 + h.k-2
    = 3^k + h.k-1 + h.k-2

    but once again am stuck as to what to do.
    Any help muchly appreciated
    Check the base case, then assume the for some k, h_k<3^k, then:

     <br />
h_{k+1}=h_k+h_{k-1}+h_{k-2}<br />

    but the sequence h_n is increasing so if h_k<3^k, then also h_{k-1}<3^k and h_{k-2}<3^k. hence:

     <br />
h_{k+1}=h_k+h_{k-1}+h_{k-2}<3 \times 3^k = 3^{k+1}<br />

    which proves the induction step, and I can leave the rest to you.

    RonL
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