There's at least two ways to do this problem. One is to note that every other term looks like n - (n-1) = 1. How many of these terms are there? Well, there are 101 - 2 + 1 = 100 terms. Hence, the sum is 100.

The other way of doing this, that I know of, is to see that you really have a telescoping series. That is, there's a -2 for the 2, a -3 for the 3, and so on. Not every number has a corresponding negative, however. To see which ones don't, let me write out just a few more terms explicitly:

2-1+3-2+4-3+5-4+6-5+…+99-98+100-99+101-100.

So we can see that the -1 in the beginning doesn't have a +1 to cancel it out. Also, the 101 doesn't have a canceling term. Hence, the sum is 101 - 1 = 100.