You might already know the sum of the numbers from 1 to n is equal to:
\(\frac{n(n+1)}{2}\)
There is a little trick to use it for an even numbered series.
We are calculating the following:
\(2 + 4 + 6 + \dots + n\)
when n is an even number. You just have to notice that every number there is divisible by 2, so you can factor a 2 out of the whole sum and when you do that you get:
\(2(1 + 2 + 3 + \dots + \frac{n}{2})\)
where n/2 is also a whole number. Now you have the sum of the numbers from 1 to (n/2) and then you just have to multiply by two. So, in your case, for the evens from 2 to 1000, you factor out the 2 and you get 2 times the sum from 1 to 500. The sum from 1 to 500 is equal to (500501)/2 but then you have to multiply this by 2 to get the right answer. So it is simply 500501.
