We can use a clever trick to solve this problem.
First, divide everything by two (just to simplify things). We'll multiply back in at the end.
We have:
1+2+3+4...+47+48+49+50
Note that we can pair up numbers starting with the lowest and the highest, so that we have 1+50, 2+49, 3+48, etc. These all add up to 51! As we have a bunch of pairs of numbers that add to 51, the average of the whole thing is the average of one of these pairs of numbers, or 51/2. But, we need to multiply back by 2 at the end, so our answer is 51.