The terms are decreasing by 3. For example, 1=4-3, -2=1/3, ...
So we have an arithmetic series starting with the number 4 and common difference -3.
So the nth term is 4-3(n-1)=4-3n+3=7-3n. If we put n=1, we get 7-3=4, the 1st term. If we put n=4 we get 7-12=-5, the 4th term. We can write out the series:
7-3, 7-6, 7-9, 7-12, ...
To sum the series we see that there are as many 7s as there are terms, so if there are n terms we have n sevens, that is, 7n.
Now we can forget the 7s for a moment and look at the rest of the terms: -3, -6, -9, -12, then -15, -18, -21, -24.
That gives us 8 terms. Now take the terms in pairs:
(-3-24)+(-6-21)+(-9-18)+(-12-15)=-27-27-27-27. If we just took the first 4 terms we'd get (-3-12)+(-6-9)=-15-15.
-27 is -3×9 and -15 is -3×5. 9 is 1 more than 8 and 5 is 1 more than 4. So -27=-3(n+1) and we have half as many pairs as we had terms, so that's n/2.
So the sum of these negative quantities is -3n(n+1)/2; then we add those 7s: 7n-3n(n+1)/2 is the sum of the series up to n terms. You can check this by adding a few terms and putting the value of n into the formula. For n=1 we get: 7-3×1×2/2=7-3=4, which is correct (just the first term).