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Math Help - Proof sequence

  1. #1
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    Proof sequence

    Can you help for this question ?
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  2. #2
    MHF Contributor

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    So: a_1= 1 and then a_n= 2a_{\lfloor n/2\rfloor}

    Start by calculating values until you see a pattern:

    For n= 2, n/2= 1 and so a_2= 2a_1= 2

    For n= 3, n/2= 1.5 so a_3= 2a_1= 2

    For n= 4, n/2= 2 so n_4= 2a_2= 4

    For n= 5, n/2= 2.5 so n_5= 2a_2= 4

    For n= 6, n/2= 3 so n_6= 2a_3= 4

    For n= 7, n/2= 3.5 so n_7= 2a_3= 4

    For n= 8, n/2= 4 so n_8= 2a_4= 8

    Conjecture: if n is a power of 2 then a_n= n. If 2^k< n< 2^{k+1} then a_n= 2^k< n.
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