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If I have two summations, $\sum_{k=1}^{n} k^2$ and $\sum_{k=1}^{n+1} k^2$ and I subtract the summation with the larger stopping point from the one with the smaller stopping point, like so

$\sum_{k=1}^{n+1} k^2 - \sum_{k=1}^{n} k^2$

will the result be $(n+1)^2$? If not, why?

McFizz
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    Yes, it will. Can you say a little about why you're unsure? – Chris Jun 16 '17 at 05:34
  • @Chris Because I'm using that for a proof for a discrete math class and it seemed a bit cheap to be able to use that. Maybe there's another mistake I'm not catching, or it's easier than I thought lol – McFizz Jun 16 '17 at 05:36
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    How do you even define $\sum_{k=1}^n$ if not by recursion and this very property? (I mean formally, that is without using "$\ldots$" anywhere) – Hagen von Eitzen Jun 16 '17 at 05:40
  • @Hagen von Eitzen good point! I didn't think of that – McFizz Jun 16 '17 at 05:41

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