# Problem #402

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 402 Let $F(z)=\frac{z+i}{z-i}$ for all complex numbers $z\not= i$, 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 $a+b$. This problem is copyrighted by the American Mathematics Competitions.
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• Reduce fractions to lowest terms and enter in the form 7/9.
• Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
• Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
• Exponents should be entered in the form 10^10.
• If the problem is multiple choice, enter the appropriate (capital) letter.
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