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Math Help - If the sum of the consecutive integers from -22 to x, inclusive, is 72

  1. #16
    Newbie Potterpie's Avatar
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    Sorry, add 23 to what?
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  2. #17
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    Quote Originally Posted by Potterpie View Post
    Sorry, add 23 to what?
    The sum of consecutive integers from -22 to 22, which is equal to 0.
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  3. #18
    Newbie Potterpie's Avatar
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    23.. =)

    I'm lost. Sorry. You don't have to try.
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  4. #19
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    Quote Originally Posted by Potterpie View Post
    If the sum of the consecutive integers from -22 to x, inclusive, is 72
    then what is the value of x?

    this question is from my SAT study guide. Its difficulty is hard.

    Thanks so much!

    72 = -22 + (-21) + .... + 0 + ..... + (x - 1) + x .... (A)

    Reverse the order:

    72 = x + (x - 1) + ..... + 0 + ..... + (-21) + (-22) .... (B)

    Add (A) and (B) together:

    144 = (x - 22) + (x - 22) + ..... + (x - 22) + (x - 22).

    There are 23 + x identical terms on the right. Therefore:

    144 = (x - 22)(23 + x).

    Solve for x (you can do that, right?) and keep the positive solution. I get x = 25.
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  5. #20
    Newbie Potterpie's Avatar
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    Yes, I can solve for x.

    Thanks. How did you know to reverse the order and add?
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  6. #21
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    Quote Originally Posted by Potterpie View Post
    Yes, I can solve for x.

    Thanks. How did you know to reverse the order and add?
    It's a trick I learned from Carl Gauss. If you want, you can read about him here: Carl Friedrich Gauss - Wikipedia, the free encyclopedia. He's a bit famous in certain circles .....
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  7. #22
    Newbie Potterpie's Avatar
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    Okay, well, thanks. Still kind of confused but I get it a little better now.
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  8. #23
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    Quote Originally Posted by mr fantastic View Post
    It's a trick I learned from Carl Gauss. If you want, you can read about him here: Carl Friedrich Gauss - Wikipedia, the free encyclopedia. He's a bit famous in certain circles .....
    He is awesome. I have his Disquisitiones Arithmeticae, very beautiful book
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  9. #24
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    Quote Originally Posted by Potterpie View Post
    Okay, well, thanks. Still kind of confused but I get it a little better now.
    Another way of doing it - which has probably been mentioned, but I can't be bothered reading through the thread again - is to realise that if x = 22, then the sum is zero. Then note that 23 + 24 + 25 = 72. So x = 25.

    No doubt that's how you're expected to do it.
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