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Pretty simple question. I always used this, because I know this holds, but to be honest I don't know why.

$$ \sum_{i=1}^n \sum_{j:j<i} \frac{1}{2} = \frac{1}{2} \sum_{i=1}^n (n-i)$$

So yeah as I said, I know this works. But I would like to have a logical explanation why I can exchange the second summation with (n-i). Thanks in advance for the answer.

nummer92
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  • If you know why it works, you would not be asking why you can replace it with n-i. – TMM Apr 17 '17 at 12:17
  • Draw the (i,j)-index area for your summation in a plane. They form a triangle (LHS). Now the RHS sums the rows of this triangle, measuring with $(n-i)$ the length of these rows. – Andreas Apr 17 '17 at 12:20
  • @TMM: No that's what bothers me. I don't know why it works. I only know that it does. If I see the leftern side of the equation, I know I can transform it to the rightern side. – nummer92 Apr 17 '17 at 12:30
  • How can you "know" that it works if you don't know why it works? That means you don't really know at all. – TMM Apr 17 '17 at 12:38
  • I was probably teached or shown this once. Probably I was also shown/teached why this holds. But I do not know that anymore. Can't you tell or explain to me why I can replace the second summation with n-i? – nummer92 Apr 17 '17 at 12:42

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