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How can I prove that $2-1+3-2+4-3+…+(n+1) - n = -1 + (n + 1)$?

I know that $2$ and $-2$ cancel out, and so do $3$ and $-3$, $4$ and $-4$, and so on, but is there a way I can prove that in such a sum, every term is cancelled out except the second term and the second to last term? (In this example, $-1$ and $n + 1$.)

Actually, is merely saying that most of the terms cancel out enough?

$n$ is a positive integer.

Léo Lam
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    https://en.wikipedia.org/wiki/Telescoping_series – Victor Sep 15 '15 at 21:54
  • @Victor thanks, it does look like my sum is a "telescoping series". That's something I had never heard! (I'm just in the first year of high school.) – Léo Lam Sep 15 '15 at 21:57
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    For this particular sum another way is to write it like this: $(2-1)+(3-2)+(4-3)+\cdots+((n+1)-n)$, and simplify the things in parentheses before summing. But in general it's very useful to know about telescoping series. – David K Sep 15 '15 at 22:02

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