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Math Help - 1^2 - 2^2 + 3^2 - 4^2 + 5^2 - ...... -100^2

  1. #1
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    1^2 - 2^2 + 3^2 - 4^2 + 5^2 - ...... -100^2

    How would I go about finding the sum of this series? Surely theres gotta be an easier way than throwing it all into the calculator. Thanks.
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  2. #2
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    Can you follow this?
    \begin{array}{rcl}<br />
 \sum\limits_{k = 1}^{100} {\left( { - 1} \right)^{k + 1} k^2 } & = & \sum\limits_{k = 1}^{50} {\left[ {\left( {2k - 1} \right)^2  - \left( {2k} \right)^2 } \right]}  \\ <br />
  & = & \sum\limits_{k = 1}^{50} {\left[ {1 - 4k} \right]}  \\ <br />
  & = & 50 - 4\frac{{\left( {50} \right)\left( {51} \right)}}{2} \\ <br />
 \end{array}<br />
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  3. #3
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    ^Thanks! It seems to work for the rest of the problem, but what did you do to get the first step on the right side?
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  4. #4
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    Hello, GeeDee!

    Another approach . . .


    We have: . S \:=\:1^2-2^2+3^2-4^2+5^2-6^2 + \cdots + 99^2 - 100^2

    . . = \;(1^2-2^2) + (3^2-4^2) + (5^2-6^2) + \cdots + (99^2-100^2)

    . . = \;(1-2)(1+2) + (3-4)(3+4) + (5-6)(5+6) + \cdots + (99-100)(99+100)

    . . = \;-1 - 2 - 3 - 4 - 5 - 6 - \cdots - 99 - 100


    This is an an arithmetic series with first term a = \text{-}1,
    . . common difference d = \text{-}1, and n = 100 terms.

    Its sum is: . S \;=\;\frac{100}{2}\left[2(\text{-}1) + 99(\text{-}1)\right] \;=\;\boxed{-5050}

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  5. #5
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    Quote Originally Posted by Soroban View Post
    Its sum is: . S \;=\;\frac{100}{2}\left[2(\text{-}1) + 99(\text{-}1)\right] \;=\;\boxed{-5050}[/size]
    Soroban, do you really care about students learning mathematics?
    I am sure that only students who what a completely ready "to handin” solution can appreciate your reply.
    May I ask if you really want anyone to learn any mathematics?
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  6. #6
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    Hello, Plato!

    What kind of silly-ass question is that?

    Are you taking lessons from Stapel and Mathman?

    I've already left one math site because of this snotty attitude.

    I taught math (successfully) for forty years. When students come to me for help,
    I have never folded my arms and said, "If you can't show me your work, GET OUT!"

    I have never believed in the Socratic approach . . . it takes far too long to DRAG
    the reasoning and answers out of each and every student. I have found that
    one clear, detailed solution can clear up the mystery for most of my students.

    Certainly, many students simply want their homework done for them. Usually,
    I can't tell from their initial posting. But if they reply with something immature like
    "Are you SURE that's the right answer?", then I know I've been "had" and I won't
    help that student again. Or I get no feedback from my solution and instead I get,
    "Okay, how about this one...?", another loser ... lesson learned.

    Do I want them learn? . . . Give me a break!
    My life has been devoted to Teaching . . . not scolding or making snide remarks.

    I give all student "the benefit of the doubt" at first. I do NOT treat them as
    academic vagrants looking for a handout. I assume they are sincere in their quest,
    whether it is a method, a hint, or a detailed explanation . . . which I assume
    they will use for a template for the rest of their assignment.

    Personally, I don't care how you teach or what you think. Just keep your little
    snipes to yourself, okay?

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  7. #7
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    I do apologize to you for not knowing your philosophy of education. But I do have a few comments and observations.
    Quote Originally Posted by Soroban View Post
    Are you taking lessons from Stapel and Mathman?
    Who are they? If they agree with me that the show and tell teaching method has never worked in mathematics, then I welcome their input.

    Quote Originally Posted by Soroban View Post
    I have never believed in the Socratic approach . . . it takes far too long to DRAG the reasoning and answers out of each and every student. I have found that one clear, detailed solution can clear up the mystery for most of my students.
    I must say, I find that a shocking statement for any teacher.
    I do respect that you think that, I infer that you a basically a trainer.
    Is that a fair statement? Does that define the different in what community colleges do (i.e. train) and what universities do?

    Quote Originally Posted by Soroban View Post
    I give all student "the benefit of the doubt" at first. I do NOT treat them as academic vagrants looking for a handout. I assume they are sincere in their quest, whether it is a method, a hint, or a detailed explanation . . . which I assume they will use for a template for the rest of their assignment.
    I find it very hard to think that giving a template is any more than training as opposed to learning.

    Again, I do apologize for not realizing that we have such a fundamental disagreement on the difference in training and learning.

    You and I clearly have a different view of mathematics education.
    I pledge to you no more from me on your training efforts
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