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Math Help - Find the largest and smallest

  1. #1
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    Find the largest and smallest

    The sum of the non-negative real numbers  s_1, s_2,..., s_{2004} is 2 and
     s_1s_2+s_2s_3+ ...+ s_{2003} s_{2004}+ s_{2004}s_1=1.
    Find the largest and smallest possible values of
    S=s_1^2+s_2^2+...+s_{2004}^2
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by perash View Post
    The sum of the non-negative real numbers  s_1, s_2,..., s_{2004} is 2 and
     s_1s_2+s_2s_3+ ...+ s_{2003} s_{2004}+ s_{2004}s_1=1.
    Find the largest and smallest possible values of
    S=s_1^2+s_2^2+...+s_{2004}^2
     \sum_{i=1}^{2400} s_i =2

    Now square:

     \left(\sum_{i=1}^{2400} s_i\right)^2 = \sum_{i=1}^{2400} s_i^2 +2\sum_{i \ne j} s_i s_j = \sum_{i=1}^{2400} s_i^2 +2 =4

    So take it from there.

    RonL
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  3. #3
    Senior Member JaneBennet's Avatar
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    Quote Originally Posted by CaptainBlack View Post
     \left(\sum_{i=1}^{2400} s_i\right)^2 = \sum_{i=1}^{2400} s_i^2 +\color{red}2\sum_{i \ne j} s_i s_j \color{black}= \sum_{i=1}^{2400} s_i^2 +\color{red}2\color{black} =4
    No. \sum_{i \ne j}{s_i s_j} is not the same as <br />
s_1s_2+s_2s_3+\ldots+ s_{2003} s_{2004}+ s_{2004}s_1<br />
. The former contains terms like s_2s_4 that are not present in the latter. Rather,

    S\ =\ s_1^2+s_2^2+\ldots+s_{2004}^2

    =\ (s_1+\ldots+s_{2004})^2-2\sum_{i \ne j}{s_i s_j}

    \leq\ (s_1+\ldots+s_{2004})^2-2(s_1s_2+s_2s_3+\ldots+ s_{2003} s_{2004}+ s_{2004}s_1)

    =\ 2^2-2\ =\ 2

    For the lower bound, the Cauchy–Schwarz inequality gives

    |s_1s_2+s_2s_3+\ldots+ s_{2003} s_{2004}+ s_{2004}s_1|\ \leq\ (s_1^2+s_2^2+\ldots+s_{2004}^2)^{\frac{1}{2}}(s_2^  2+s_3^2+\ldots+s_{2004}^2+s_1^2)^{\frac{1}{2}}<br />

    so S\ \geq\ 1.
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