Finite Arithmetic Series

There exists a unique strictly increasing arithmetic sequence $\{a_i\}_{i=1}^{100}$ of positive integers such that $$a_1+a_4+a_9+\cdots+a_{100}=\text{1000},$$ where the summation runs over all terms of the form $a_{i^2}$ for $1\leq i\leq 10$. Find $a_{50}$.

(Source: NIMO 18)