jmc

The difference between consecutive squares is $(x+1{)}^2-x^2=2x+1$, which means that all squares above $50^2=2500$ are more than $100$ apart. Then the first $26$ sets ($S_0,\cdots S_{25}$) each have at least one perfect square. Also, since $316^2<100000$ (which is when $i=1000$), there are $316-50=266$ other sets after $S_{25}$ that have a perfect square. There are $1000-266-26=\boxed{708}$ sets without a perfect square.