Mod on Sequences

While solving a problem at one of the Programming Practice sites, I came to know this about sum of a sequence. I wonder how much lacking do I have in Maths.

If, for example, 1+2+3+.......+(n-1)+n is the sequence that you want the sum of then, what we do is

add all the numbers from 1 to n. But this has another method. Whenever the sum crosses n during the the summation process you can take it as sum%n.

Which means, (1+2+3+.........+n)%n = (sum+1)%n + (sum+2)%n + ...... + ( sum+n)%n.

I didn't get it from any book so not sure if it's wrong or right. If someone finds it a mistake please comment.