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Math Help - Couple of similar proofs

  1. #1
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    Couple of similar proofs

    Let x1,x2,..., xn be positive real numbers. Prove that

    n^2 <= (x1 + x2 + ... + xn ) ( 1/x1 + 1/x2 + ... + 1/xn )

    and

    (x1 + x2 + ... + xn)/(sqrt(n)) <= sqrt( x1^2 + x2^2 +...+ xn^2)
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  2. #2
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    Still not able to solve the second one, any help appreciated.

    I was able to solve the first one using induction and the lemma from another thread i asked about that (x/y) + (y/x) > 2

    Using induction, you can factor the sequence of k + 1 in to the sequence up to k, and (x/y) + (y/x) for each where x = x1, x2,...,xk and y = x(k+1). Since there are k of those factors and each is greater than 2, we have the first two factors are great than k^2 + 2k. Also, when factoring you get a factor of 2, so with K^2 + 2k + 2 > K^2 + 2k + 1 = (K+1)^2, the theorem is proved by induction.
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  3. #3
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    Nvm, the second question is quite easy.

    Let a_n = x1 + x2 + ... + xn
    let b_n = 1 + 1 + ... + 1 = n

    Then the inequality holds from the Cauchy-Swartz inequality using a little simplification
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