2002 AIME I Question 12

Let $F(z) = \frac{z + i}{z - i}$ for all complex numbers $z \ne i,$ and let $z_n = F(z_{n-1})$ for all positive integers $n.$ Given that $z_0 = \frac{1}{137} + i$ and $z_{2002} = a + bi,$ where $a$ and $b$ are real numbers, find $a + b.$