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Math Help - Summation formula

  1. #1
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    May 2007
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    Summation formula

    Hello

    Prove that the sum of k^2 from k=1 to n is equal to:

    (n^3)/3 + (n^2)/2 + n/6

    The book provided a hint: k^3 -(k -1)^3 = 3k^2 -3k + 1.
    But I don't know how to use it.

    Thanks in advance
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  2. #2
    Member
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    Jan 2008
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     (v+1)^{3} - v^{3} = 3v^{2} + 3v+1

    Write this equation down for  v = 0..n and add them.

    Then  (n+1)^{3} = 3S_2 + 3S_1+n+1 and  3S_{2} = (n+1) \left[(n+1)^{2} - 1 -\frac{3}{2}n \right] = \left(n+1\right)\left(n^{2} + \frac{1}{2}n \right)

     S_2 = \frac{1}{6}n(n+1)(2n+1)
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