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Math Help - Airthmetic-geometric means problem?

  1. #1
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    Airthmetic-geometric means problem?

    i have a question from an analysis paper that i think can be solved using the arithmetic-geometric means inequality. This is the question:

    show that if a_1, a_2, a_3, ... , a_n are positive, such that

    (a_1)(a_2)(a_3)...(a_n)=1

    then

    (1+a_1) (1+a_2) (1+a_3) ... (1+a_n) ≥ 2^n


    it makes sense because using the AM-GM inequality, 'on average' the a_i are greater than 1, so 'on average you have something greater than
    (1+1) (1+1) ...(1+1) = 2^n

    but i don't know how to prove it rigourously.
    can anyone help?
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  2. #2
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    This is true for all x & y: x^2  + y^2  \ge 2xy. Therefore in your case, 1 + a_j  \ge 2\sqrt {a_j } .

    So it follows that \prod\limits_{j = 1 }^n {\left( {1 + a_j } \right)}  \ge \prod\limits_{j = 1}^n {2\sqrt {a_j } }  = 2^n \sqrt {\prod\limits_{j = 1}^n {a_j } } .
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