The sum of integers between 3 and 1000 means the sum of multiples of 7 between 3 and 1000. (it does not matter for this problem whether we include or exclude the endpoints because neither are multiples of 7 :P)
3<7n<1000 divide both sides by 7
3/7<n<142.8 so n varies from 1 to 142
The first term is 7 and the last term is 994, and this set consists of (142-1+1)=142 terms.
The sum of any arithmetic sequence is the average of the first and last terms times the number of terms. In this case:
s=142(7+994)/2
s=71071
The sum of multiples of 7 between 3 and 1000 is 71071.