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Math Help - 1 + 2 + 3 + ... + n

  1. #1
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    1 + 2 + 3 + ... + n

    I'm reading What is mathematics? and I got confused in page twelve where it says about Arithmetical progression.

    I don't understand the "Adding the number ( r + 1)" thing.

    This is the page:

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  2. #2
    Senior Member abhishekkgp's Avatar
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    the image you have posted is not visible probably due to some site problem. If you can tell me in which section of the book you have this then i can help.
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  3. #3
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    We'll call this sum S.

    We can write S = 1 + 2 + 3 + ... + (n - 2) + (n - 1) + n.

    We can also write it as S = n + (n - 1) + (n - 2) + ... + 3 + 2 + 1.

    If we add the sums written in these two forms together we get

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

    2S = (n + 1) + (n + 1) + (n + 1) + ... + (n + 1) + (n + 1) + (n + 1)

    2S = n(n + 1)

    S = n(n + 1)/2.
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  4. #4
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    I tried to upload it again abhishekkgp.

    I'm trying to understand it Prove It.
    Last edited by GustavoB; April 17th 2011 at 07:26 AM.
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  5. #5
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    Hello, GustavoB!

    I'm reading What is mathematics? and I got confused in page twelve
    where it says about Arithmetical progression.

    I don't understand the "Adding the number (r + 1)" thing.

    It sounds like a Proof by Induction.

    Are you familiar with it?

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  6. #6
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by GustavoB View Post


    I tried to upload it again abhishekkgp.

    I'm trying to understand it Prove It.
    i guess soroban's post will solve your query. prove-it has proved it using a different method.
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  7. #7
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    The Best Method Is Prove-It 's method

    Induction is a second alternative
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  8. #8
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    What sounds strange to me is that he adds a (r + 1) to both sides of the equation, then it became r(r + 1) + 2(r+1) and so on.

    I don't understand this process.

    Prove it already tried to explain me but i still can't see why to do a Sum S.

    This page is a part of the book where the author tries to explain Proof by induction.
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  9. #9
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by GustavoB View Post
    What sounds strange to me is that he adds a (r + 1) to both sides of the equation, then it became r(r + 1) + 2(r+1) and so on.

    I don't understand this process.

    Prove it already tried to explain me but i still can't see why to do a Sum S.

    This page is a part of the book where the author tries to explain Proof by induction.

    i think it must have been this way in the book
    define:
    f(r) = 1+2+3+...+r
    to prove that f(r)= r*(r+1)/2

    induction:
    f(1) is true
    assume f(r) is true.
    then f(r+1) = f(r) +(r+1). do you see why?
    so f(r+1)= r(r+1)/2 + (r+1)
    so f(r+1)= (r+1)[r/2 + 1]
    so f(r+1)= (r+1)(r+2)/2
    so f(r+1) is also true.
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