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Math Help - Induction

  1. #1
    Senior Member tukeywilliams's Avatar
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    Induction

    Prove that  \frac{1}{n} \sum_{i=1}^{n} x_i \geq \left(\prod_{i=1}^{n} x_i \right)^{\frac{1}{n}} for positive integers  n and positive real numbers  x_i .

    I dont think I can do this directly with induction on  n . I let  n = 2^{m} for  m \geq 0 . So the induction hypothesis is the following:  \frac{1}{2^{m}} \sum_{i=1}^{2^{m}} x_i \geq \left(\prod_{i=1}^{2^{m}} x_i \right)^{\frac{1}{2^{m}}} . In other words, I have to prove that  P(k+1) \Rightarrow P(k) . I would use strong induction then?

    Any help is appreciated. Thanks.
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