I first heard about this kind of problem in connection with Fred Gauss,
who went on to pioneer symmetry arguments in Physics.
I think he did something with series like this too.
In that context it was only the numbers from 1 to 10 and his teacher wanted some time to rest while the students busied themselves with mindless addition. Gauss (in pre-calculator days) produced the answer instantly - much to the teacher's chagrin.
It was a collegue who told me this, but I have never verified it.
In this more advanced problem he would have noticed that
0 + 100 = 100,
1 + 99 = 100
49 + 51 =100
This happens 50 times with a 50 left over.