When is the Probability One-Half?

Let $b$ and $w$ be the number of black and white marbles in the urn, respectively. Then the probability of two distinct colors being picked is $\dfrac b{b+w}\cdot\dfrac w{b+w-1}+\dfrac w{b+w}\cdot\dfrac b{b+w-1}=\dfrac{2bw}{(b+w)(b+w-1)}.$ Setting this equal to $\tfrac12$ and cross multiplying gives $\begin{aligned}4bw&=(b+w)(b+w-1)\\&=b^2+2bw+w^2-b-w\\\implies b+w&=b^2-2bw+w^2=(b-w)^2.\end{aligned}$ It is easy to see that since $b-w$ and $b+w$ are of the same parity, for any value of $b-w$ there will exist an integer pair $(b,w)$ that works. The only exception to this is $b-w=1$ , as this gives $(b,w)=(1,0)$ which is bad. Hence $k_i=(i+1)^2$ , so $\sum_{i=1}^{100}k_i\equiv\sum_{i=1}^{101}i^2-1\equiv \dfrac{101\times 102\times 203}6-1\equiv\boxed{550}\pmod{1000}.$