# Summations: Can Someone Explain The Following Solution to a Summation Problem?

I have the following solution to a problem that I'm attempting to understand but I cannot find a rule online which explains it. Can someone please explain where the i comes from in the following summation?

$X=\sum\limits_{i=1}^{n-1}\sum\limits_{j=i}^{2n+1}(1)$

(In the following line I understand where the $(2n + 2)$ comes from because since 1 is being subtracted in the index you must add 1 to the variable. But i do not understand why the i is added why is it not just $\sum\limits_{i=1}^{n}(2n + 2 + 1)$ which is what i arrived at in my own solution?)

$=\sum\limits_{i=1}^{n-1}(2n+1-i+1) = \sum\limits_{i=1}^{n-1}(2n+2-i)$

$=(n-1)(2n+2) - \sum\limits_{i=1}^{n-1}(i)$

(please also explain how the second term is derived? Why is is not the normal arithmetic sequence sum of: $n(n + 1) / 2$ ?)

$=(n-1)(2n+2) - (n-1)\frac{1+(n-1)}{2}$

The rest of the solution is trivial simplification that I understand. the lines which begin with = are the actual solution parenthetical statements are my personal thoughts / questions

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can you possibly explain further as i do not understand why 5 and 10 must be considered in finding a solution to the problem. –  user17321 Nov 17 '13 at 1:17