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Math Help - Sums of Series

  1. #1
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    Sums of Series

    Hi, well got my exam tomorrow and this is the only topic area I struggle with soo much, a past paper i'm trying here asks this qustion:

    It is given that Sn=\sum(3r^2-3r+1)

    Use the formulae for \sum(r^2) and \sum(r) to show that Sn=n^3.

    I know that I treat the +1 as n but I really dont know where I should take out factors or when I should expand etc. Also I get the formulas for the r squared and r so im aware of what they are.

    Each time Ive tried Ive managed to get to either n^3-1 or n(n^2+1)

    I would be really greatful if someone could explain what process I should do to get the answer.

    Thankyou !
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  2. #2
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    Quote Originally Posted by NathanBUK View Post
    Hi, well got my exam tomorrow and this is the only topic area I struggle with soo much, a past paper i'm trying here asks this qustion:

    It is given that Sn=\sum(3r^2-3r+1)

    Use the formulae for \sum(r^2) and \sum(r) to show that Sn=n^3.

    I know that I treat the +1 as n but I really dont know where I should take out factors or when I should expand etc. Also I get the formulas for the r squared and r so im aware of what they are.

    Each time Ive tried Ive managed to get to either n^3-1 or n(n^2+1)

    I would be really greatful if someone could explain what process I should do to get the answer.

    Thankyou !
    You probably went about it right but made some small error somewhere along the way. Here are my steps.

    \sum_{r=1}^nr^2=\frac{n(n+1)(2n+1)}{6}

    \sum_{r=1}^nr=\frac{n(n+1)}{2}

    So

    \sum_{r=1}^n\left(3r^2-3r+1\right)= \sum_{r=1}^n3r^2-\sum_{r=1}^n3r+\sum_{r=1}^n1 =

     3\sum_{r=1}^nr^2-3\sum_{r=1}^nr+\sum_{r=1}^n1 = 3\left(\frac{n(n+1)(2n+1)}{6}\right)-3\left(\frac{n(n+1)}{2}\right)+n =

    \frac{n(n+1)(2n+1)}{2}-\frac{3n(n+1)}{2}+n = \left(\frac{1}{2}\right)(n)(n+1)(-3+2n+1) + n =

    \left(\frac{1}{2}\right)(n)(n+1)(2n-2) + n =

    n(n+1)(n-1) + n = n(n^2-1) + n = n(n^2 - 1 + 1) = n(n^2) = n^3
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  3. #3
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    Ohh thankyou ! Lets just hope one too nasty doesnt come up on my test
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