Hi Krista.

In an arithmetic series, every term is separated from its neighbours
by the same difference.
So given an arithmetic sequence that starts with the first value A and
has a common difference of D, the first five terms are:
1st term: A
2nd term: A + D
3rd term: A + 2D
4th term: A + 3D
5th term: A + 4D

So the first five sums of the terms are:
1st sum: A
2nd sum: 2A + D
3rd sum: 3A + 3D
4th sum: 4A + 6D
5th sum: 5A + 10D

You know that the sum of the first 5 numbers is 500, so:
500 = 5A + 10D

and that the sum of the first 4 numbers is -8, so
-8 = 4A + 6D

Now you have two equations with two unknowns.

You can solve this and find the actual values of A and D. From those,
you can quickly write the first three numbers in the series and add
them up, or simply use the formula for the third sum: 3A + 3D

Hope this helps,
Stephen La Rocque.