Also got 20200 but my approach was less intuitive than Dookie's.
If you look at pattern of increase b/w first 5 positive integer, you get:
Now, the pattern of increase b/w first 5 even integer is:
If sum of first n positive integer is n(n+1)/2, then sum of first even n positive integer is 2n(n+1)/2 = n(n+1)
I looked for diff. b/w first 150 even integers (2-300 inclusive) and diff. b/w first 50(2-98 inclusive) even integers. I get 150(151) - 50(51) = 20200. Took me about 2 min.