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Math Help - summation question

  1. #1
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    summation question

    Maple says that

    summation question-double-sum-abs-value.jpg

    why is each of the|Xi-Xj| multiplied by two?

    Is it that |Xi - Xj| =|Xj - Xi|?

    I had not realized that double summation multiplies sums every combination of Xi and Xj
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  2. #2
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    Re: summation question

    Quote Originally Posted by kingsolomonsgrave View Post
    Is it that |Xi - Xj| =|Xj - Xi|?
    Yes. There are nine ordered pairs (i, j), and three of them (with equal elements) give rise to |X_i - X_i| = 0. The other six are split into three pairs because |X_i - X_j| =|X_j - X_i|.

    If you want just |X_1-X_2|+|X_1-X_3|+|X_2-X_3|, you can denote it by \sum_{1\le i<j\le 3}|X_i-X_j|.
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  3. #3
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    Re: summation question

    summation question-9-oct-13-2.png
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  4. #4
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    Re: summation question

    On line 4, you keep i=1 for all three terms. On line 5, you keep i=2 for all three terms. Then on line 6, you keep i=3 for all three terms.

    It should read (beginning on line 4):
    \begin{align*}= & \hspace{1em}\hspace{3pt} |x_1-x_1| + |x_2-x_1| + |x_3-x_1| \\ & + |x_1-x_2| + |x_2-x_2| + |x_3-x_2| \\ & + |x_1-x_3| + |x_2-x_3| + |x_3-x_3| \\ = & \hspace{1em}\hspace{3pt} 0 + |x_2-x_1| + |x_3-x_1| \\ & + |x_1-x_2| + 0 + |x_3-x_2| \\ & + |x_1-x_3| + |x_2-x_3| + 0 \\ = & \, \left(|x_1-x_2| + |x_2-x_1|\right) + \left(|x_1-x_3| + |x_3-x_1|\right) + \left(|x_2-x_3| + |x_3-x_2|\right) \\ = & \, 2\left(|x_1-x_2| + |x_1-x_3| + |x_2-x_3|\right)\end{align*}
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  5. #5
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    Re: summation question

    Quote Originally Posted by ibdutt View Post
    Click image for larger version. 

Name:	9 Oct 13 - 2.png 
Views:	4 
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ID:	29421
    Sorry i made some typing mistake, it shd be as under
    summation question-10-oct-13.png
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  6. #6
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    Re: summation question

    Yes, that is correct.
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