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Homework Help: Ratio of partial sum to total sum

  1. Jun 23, 2010 #1

    txy

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    1. The problem statement, all variables and given/known data

    Given n real numbers [tex]x_1, x_2, \dotsb , x_n[/tex] which satisfy [tex]0 \leq x_1 \leq x_2 \leq \dotsb \leq x_n[/tex],
    show that
    [tex]\frac{x_1 + x_2 + \dotsb + x_k}{x_1 + x_2 + \dotsb + x_n} \leq \frac{k}{n}, \forall 1 \leq k \leq n[/tex].

    2. Relevant equations



    3. The attempt at a solution

    If [tex]x_1 = x_2 = \dotsb = x_n[/tex], then
    [tex]\frac{x_1 + x_2 + \dotsb + x_k}{x_1 + x_2 + \dotsb + x_n} = \frac{k(x_1)}{n(x_1)} = \frac{k}{n}[/tex].

    If they are not all equal, suppose [tex]x_1 \neq x_2[/tex], then everything except [tex]x_1[/tex] would be strictly positive. Then I don't know how to continue. I can't seem to get a nice inequality coming out.

    Edit: I just realised that by moving the terms around in the above inequality, I get
    [tex]\frac{x_1 + x_2 + \dotsb + x_k}{k} \leq \frac{x_1 + x_2 + \dotsb + x_n}{n}[/tex]
    This is like saying that the mean of a set of numbers increases when even bigger numbers are added to the set. This seems intuitive enough, but I haven't figured out how to prove this.
     
    Last edited: Jun 23, 2010
  2. jcsd
  3. Jun 23, 2010 #2

    txy

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    I think I've solved it.
    Let
    [tex]\overline{X}_k = \frac{1}{k}\left(x_1 + x_2 + \dotsb + x_k\right)[/tex]
    Then
    [tex]x_1 + x_2 + \dotsb + x_k = k\overline{X}_k[/tex]
    Because [tex]\overline{X}_k[/tex] is the mean, we have
    [tex]\overline{X}_k \leq x_k \leq x_{k+1} \leq \dotsb \leq x_n[/tex]
    So
    [tex]x_1 + x_2 + \dotsb + x_k + x_{k+1} + x_{k+2} + \dotsb + x_n
    = k\overline{X}_k + x_{k+1} + x_{k+2} + \dotsb + x_n
    \geq k\overline{X}_k + \overline{X}_k + \overline{X}_k + \dotsb + \overline{X}_k
    = n\overline{X}_k[/tex]
    And therefore
    [tex]\frac{x_1 + x_2 + \dotsb + x_k + x_{k+1} + x_{k+2} + \dotsb + x_n}{n} \geq \overline{X}_k = \frac{x_1 + x_2 + \dotsb + x_k}{k}[/tex]
     
    Last edited: Jun 23, 2010
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