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Thread: Induction

  1. #1
    Senior Member tukeywilliams's Avatar
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    Induction

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

    I dont think I can do this directly with induction on $\displaystyle n $. I let $\displaystyle n = 2^{m} $ for $\displaystyle m \geq 0 $. So the induction hypothesis is the following: $\displaystyle \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 $\displaystyle P(k+1) \Rightarrow P(k) $. I would use strong induction then?

    Any help is appreciated. Thanks.
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