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Math Help - Geometric mean and arithmetic mean by induction

  1. #1
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    Geometric mean and arithmetic mean by induction

    I need to prove that G_(2^n) <= A_(2^n) by using induction on n.
    G_n = geometric mean and A_n is arithmetic mean.

    I have a hint which says let m be such that 2^m >= n and set a_(n+1) = a_(n+2) = ... = a_(2^m) = A_n

    I don't have idea how to apply this hint on this problem.

    Please help me out...
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  2. #2
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    Re: Geometric mean and arithmetic mean by induction

    The hint pertains to the following part of the proof, where you generalize the inequality from n of the form 2k to arbitrary n. It is not used in the claim you wrote.

    The induction on n is pretty straightforward. You prove the claim for n = 2 and assume it for n = k. In proving it for n = k + 1, you use the induction hypothesis as well as the base case (for n = 2).

    Please post here if you are still having difficulties.
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