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Math Help - Summation

  1. #1
    Member helloying's Avatar
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    Summation

    Show that  \sum (-1)^{r+1} r^2 = -n(2n+1)


    summation is from r=1 to 2n . sry i dont know how to type

    Thx
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  2. #2
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    Quote Originally Posted by helloying View Post
    Show that  \sum (-1)^{r+1} r^2 = -n(2n+1)


    summation is from r=1 to 2n . sry i dont know how to type

    Thx
    hi

    \sum^{2n}_{r=1}(-1)^{r+1} r^2 will generate

     \sum^{2n}_{r=1}(-1)^{r+1} r^2

    =1-4+9-16+25-36+...

    =1^2-2^2+3^2-4^2+...+(2n-1)^2-(2n)^2

    =(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n-1)

    =-3-7-...-(4n-1)

    which is an AP with first term -3 , last term -(4n-1)

    S_n=\frac{n}{2}(-3-(4n-1))

    simplify this and you are done.
    Last edited by mathaddict; May 27th 2010 at 11:37 PM. Reason: algebraic error
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  3. #3
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by mathaddict View Post
    hi

    \sum^{2n}_{r=1}(-1)^{r+1} r^2 will generate

     \sum^{2n}_{r=1}(-1)^{r+1} r^2

    =1-4+9-16+25-36+...

    =1^2-2^2+3^2-4^2+...+(2n)^2-(2n-1)^2

    =(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n+1)

    =-3-4-...-(4n-1)

    which is an AP with first term -3 , last term -(4n-1)

    S_n=\frac{n}{2}(-3-(4n-1))

    simplify this and you are done.
    I didn't quite follow this.

    How did you get =1^2-2^2+3^2-4^2+...+(2n)^2-(2n-1)^2?

    When I do this I come up with =1^2-2^2+3^2-4^2+...-(2n)^2 +(2n-1)^2, because (-1)^{2n+1} {2n}^2 = -{2n}^2.
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  4. #4
    Member helloying's Avatar
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    Quote Originally Posted by oldguynewstudent View Post
    I didn't quite follow this.

    How did you get =1^2-2^2+3^2-4^2+...+(2n)^2-(2n-1)^2?

    When I do this I come up with =1^2-2^2+3^2-4^2+...-(2n)^2 +(2n-1)^2, because (-1)^{2n+1} {2n}^2 = -{2n}^2.
    yep i realise i dont get it. i agree with oldguynewstudnet.and also in the part <br />
=(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n+1)<br />

    it shd be -3-7 and not -3-4
    Last edited by Plato; May 24th 2010 at 04:47 AM.
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  5. #5
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    Quote Originally Posted by helloying View Post
    yep i realise i dont get it. i agree with oldguynewstudnet.and also in the part =(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n+1)
    " alt="
    =(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n+1)
    " />

    it shd be -3-7 and not -3-4
    yup , some algebraic error here and there . Edited .
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