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Math Help - Verifying inequalities with induction

  1. #1
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    Verifying inequalities with induction

    Hey, I've been having trouble with these two inequalities. I did the basis step but I'm having trouble proving for n+1.

    1) for n = 1, 2.. and a(i) are positive numbers

    (a_{1}*a_{2}...*a_{2^n})^\frac{1}{2^n} \leq \frac{a_{1} + a_{2} + ... + a_{2^n}}{2^n}

    revised (cartesian product cardinality = cardinality of each set multiplied)
    2) \mid X_{1} \times X_{2} ... \times X_{n} \mid  =  \mid X_{1} \mid * \mid X_{2} \mid ... \mid X_{n} \mid

    Could you please explain how you arrive to each conclusion? Thanks in advance.
    Last edited by Projectt; September 28th 2009 at 05:17 PM.
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  2. #2
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    \frac{x_1+x_2+\cdots+x_{2^k}+x_{2^k+1}+\cdots+x_{2  ^{k+1}}}{2^{k+1}} }\\ &=& \frac{( \frac{x_1+x_2+\cdots+x_{2^k}}{2^k} ) + ( \frac{x_{2^k+1}+x_{2^k+2}+\cdots+x_{2^{k+1}}}{2^k}  )} {2}
    Last edited by Projectt; September 29th 2009 at 02:59 PM.
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