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Math Help - elem#theo - induction + Bertrand's

  1. #1
    Member cassiopeia1289's Avatar
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    elem#theo - induction + Bertrand's

    OK: p_{n} denotes the nth prime. For n>3, show that p_{n} < p_{1} + p_{2} + ... + p_{n-1}
    the hint was to use induction and Bertrand's conjecture

    ok: so the answer is in the book, but its not helpful b/c I don't get one jump

    so they do this:
    So for n=4, p_{4} = 7 < 2+3+5, which is true.
    Assume for some k, p_{k} < p_{1} + p_{2} + ... + p_{k-1}
    By Bertrand, there exists a prime between p_{k} and 2p_{k}

    Here is where I'm lost: they go from there to p_{k+1} \leq 2p_{k} < (p_{1} + p_{2} + ... + p_{k_{1}}),
    thus proving that statement is true for k+1

    How? *so confused*
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  2. #2
    Member cassiopeia1289's Avatar
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    anyone? please - kinda urgent ...
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  3. #3
    Member cassiopeia1289's Avatar
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    HELP!!!
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    by our inductive assumption,  p_{k} < p_{1} + ... + p_{k-1}

    so add  p_{k} to both sides:

     p_{k} + p_{k} < p_{1} + ... + p_{k}

    which is the same thing as  2p_{k} < p_{1} + ... + p_{k}

    and we know  p_{k+1} \leq 2p_{k} by the conjecture, so...

     p_{k+1} < p_{1} + ... + p_{k}
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  5. #5
    Member cassiopeia1289's Avatar
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    Ah - thank you!
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