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Thread: Sequences.

  1. #1
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    Exclamation Sequences.

    The sum of n terms of the sequence 1,8,27,64,125,216.... is ________? Please, provide the explanation as well on how you arrived on the result. I know the answer is {n^2x(n+1)^2}/4 but, I simply can't figure out how it came. Any help will be appreciated. If anyone finds a different answer and can prove it, even it will work. I just want to know HOW IT CAME.
    Thanks to all.
    Last edited by mr fantastic; Feb 24th 2011 at 07:53 PM. Reason: Title.
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  2. #2
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    Quote Originally Posted by ashishgaurav View Post
    The sum of n terms of the sequence 1,8,27,64,125,216.... is ________?
    That sequence is just $\displaystyle a_n=n^3$.
    So find $\displaystyle \sum\limits_{k = 1}^n {k^3 }. $
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  3. #3
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    look at the sequence of partial sums ...

    $\displaystyle 1 , 9 , 36 , 100 , 225 , ... $

    $\displaystyle 1^2 , 3^2 , 6^2 , 10^2 , 15^2 , ...$

    note that the sequence $\displaystyle 1 , 3 , 6 , 10 , 15 , ...$ is the sequence of triangular numbers whose nth term is

    $\displaystyle \dfrac{n(n+1)}{2}$
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  4. #4
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    Google "sum of cubes"
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  5. #5
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    we can prove it by mathematical induction.
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