The first and second terms define the sequence, so let's call them a and b.

The third term would be a+b.
The fourth term would be a+(a+b) = a+2b.
The fifth term would be (a+b)+(a+2b) = 2a+3b.

The maximum possible value of the first term is when the second term is 1 (because it has to be a positive integer), so 2a = 2004-3. Therefore, the first term must be 1001.5. That's not allowed because it isn't an integer, so let's try when b is 2. Now 2a = 2004-6=1998, so a =999.

Check: 999, 2, 1001, 1003, 2004. It works!