3. Given an arithmetic sequence $a_{1}, a_{2}, \cdots, a_{1000}$, the sum of the first 100 terms is 100, and the sum of the last 100 terms is 1000, then $a_{1}=$
$$
S_{100}=100, S_{1000}-S_{900}=1000 \Rightarrow S_{200}-S_{100}=200 \Rightarrow d=\frac{1}{100} \text {, }
$$

So $S_{100}=100 a_{1}+\frac{100 \times 99}{2} d=100 \Rightarrow a_{1}=\frac{101}{200}$.