(N-1) + (N-2) +...+ 2 + 1 is a sum of N-1 items. Now reorder the items so, that after the first comes the last, then the second, then the second to last, i.e. (N-1) + 1 + (N-2) + 2 +... The way the items are ordered now you can see that each of those pairs is equal to N (N-1+1 is N, N-2+2 is N). Since there are N-1 items, there are (N-1)/2 such pairs. So you're adding N (N-1)/2 times, so the total value is N*(N-1)/2.
