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Inequality problem

  1. Sep 30, 2008 #1
    1. The problem statement, all variables and given/known data
    Let c1, c2, c3, …, c2009 be a sequence of real numbers such that |cn – cn+1| < 1 for 1 < n <2008. Show that:
    | c1+c2+…+c2009 c1+c2+…+c2008 |
    |2009............................2008...............|< ½

    2. Relevant equations

    See above

    3. The attempt at a solution

    Well I simplified by cross-multiplying, getting
    2008(c1 + c2 + ... + c2009) - 2009(c1 + c2 + ... + c2008)

    Which thus gives us
    -c1 - c2 - ... - c2008 + 2008c2009

    I want to use |cn - cn+1| < 1 but I then noticed that's only for 1<n<2008 (is this a parameter that'll affect us?) and then also that bracketing gives us

    -(c1+c2+...+c2008 - 2008c2009).

    I just don't know the next step now. Any pointers would be appreciated!
  2. jcsd
  3. Oct 1, 2008 #2
    Um, can no-one help me? I'm sure it's not that hard, I'm just missing the next step. Sorry to bump it, but it really is bugging me.
  4. Oct 1, 2008 #3


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    Staff Emeritus
    Science Advisor

    You have completely left out the right side of your inequality. What will that be after "cross multiplying"?
  5. Oct 1, 2008 #4

    so could we then change it to -c-c2-...-c2008 <2017036 - 2008c2009
    and dow something with that?
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