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Math Help - inequality relating to arithmetic and geometric means

  1. #1
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    inequality relating to arithmetic and geometric means

    Hi I am having trouble proving the following inequality:
    Let n be a natural number.
    n^{\frac{1}{n}} + n^{\frac{1}{(n+1)}} + ... + n^{\frac{1}{(2n - 1)}} \geq n^{\sqrt[n]{2}}
    The book I found this problem in suggests the use of these mean values:
    A(x_1,x_2,...,x_n) = \frac{x_1+x_2+...+x_n}{n}
    and
    G(x_1,x_2,...,x_n) = \sqrt[n]{x_1x_2...x_n}
    I have tried everything but cant get it to work!
    Any suggestions?
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  2. #2
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    Sep 2009
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    A previous section before this problem has this fact:
    If u,v are positive real numbers and n1, n2 are fixed natural numbers, then
    \sqrt[n1 + n2]{u^{n1}v^{n2}} \leq \frac{n1(u) + n2(v)}{n1 + n2}
    and thus for any two real numbers a and b, where a + b = 1, we have u^av^b\leq au +bv

    I tried using these facts as well but have still not gotten anywhere.
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