I apologize for the hard to read text - I'm not sure how to enter summations.. I have a double summation that I don't know how to solve ---
Summation of i = 1 to n (outer summation)
Summation of j = i + 1 to n (inner summation)
Terms being summed: j - i

Any help would be appreciated! Thank you.

Question

I apologize for the hard to read text - I'm not sure how to enter summations..  I have a double summation that I don't know how to solve --- 
Summation of i = 1 to n  (outer summation)
Summation of j = i + 1 to n (inner summation)
Terms being summed: j - i

Any help would be appreciated!  Thank you.

anonymous · Answer

$$\sum_{i=1}^{n}\sum_{j=1+i}^{n} j-i$$

anonymous · Answer

Thank you for putting this into the proper notation for me..  How do I go about solving this?  Specifically, i'm not sure what to do with the i term, since it depends on the outer summation.  It has been a long time since I have taken a class that describes how to solve summations, and I am at a loss as to what to do..

Thanks!

anonymous · Answer

I have started by manually expanding it to see if there is a pattern.
i=1 : (2-1)+(3-1)+(4-1)+(5-1)+...+(n-1)  which simplifies to
         1+2+3+4+...+(n-1)
i=2 :            (3-2)+(4-2)+(5-2)+...+(n-2)  which simplifies to
          1+2+3+...+(n-2)

So it counts through each iteration up to n-i
Now analyze how it ends...
when j=n (and i=n-1), the last loop is executed... j-i = n-(n-1) = 1
So... each i iteration counts up from one to n-i and each n-i counts down to 1.
Find the pattern.

I don't have it yet, but I wrote a program for it, so now I'm playing with it.

asnaseer · Answer

Perform the inner sum first.

asnaseer · Answer

i.e. use formula for arithmetic progression to calculate this first:$$\sum_{j=i+1}^nj-i=?$$

asnaseer · Answer

@nightjars Do you know the formula for the sum of an arithmetic progression?

anonymous · Answer

I don't know the formula for arithmetic progression, but I think I see 1, 2, 3, 4, 5 ... n as the result of the inner summation, since j will always start at (i + 1) and run through n, and i is subtracted from j in the summation...  Am I on the right track..?

anonymous · Answer

I'm re-inventing the wheel here... I think that @asnaseer is on the right track.

asnaseer · Answer

You should learn (and use) the formula. If you are not familiar with it, then I suggest you first look here: http://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html Then come back to this problem. Let me know if you still require more help after looking at that web page.