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Change of summation

  1. Aug 14, 2016 #1
    1. The problem statement, all variables and given/known data

    Using change of summation index show that:

    $$\sum_{k=1}^{n} (k + 1)^3 - \sum_{k=1}^{n} (k-1)^3 = (n + 1)^3 + n^3 - 1$$


    Hence show that:

    $$\sum_{k=0}^{n} k^2 = \frac{n}{6} (n + 1)(2n + 1)$$


    2. The attempt at a solution

    For the first part I changed the summation index like this:

    $$\sum_{k=2}^{n+1} k^3 - \sum_{k=0}^{n-1} k^3 = (n + 1)^3 + n^3 - 1$$

    Clearly when you get rid of the terms that are common to both summations you are left with the right hand side of the equation.

    For the second part, I can prove it by induction but don't see how it's related to the first part of the question.

    So far I've done this:

    $$\sum_{k=1}^{n} (k + 1)^3 - \sum_{k=1}^{n} (k-1)^3 = \sum_{k=1}^{n} [(k + 1)^3 - (k-1)^3] = \sum_{k=1}^{n} 6k^2 + 2$$

    But not really sure where I'm going with this.

    Any help appreciated.
     
  2. jcsd
  3. Aug 14, 2016 #2

    blue_leaf77

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    $$ \sum_{k=1}^{n} (6k^2 + 2) = (n + 1)^3 + n^3 - 1$$
    then solve for ##\sum_{k=1}^{n} k^2##.
     
  4. Aug 14, 2016 #3
    Ah of course.

    Thanks for that.
     
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