The powers of the <imaginary unit> have period four. Hence
$$
\sum_{a=0}^{200}i^a
=50(1+i-1-i)+i^{200}=1,
$$
while
$$
\prod_{b=1}^{50}i^b=i^{1+2+\cdots+50}=i^{1275}=i^3=-i.
$$
Thus $x+iy=1-i$, so
$$
\boxed{xy=-1}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 1 / 1C / a / Solution

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The imaginary unit is the complex number $i$ satisfying $i^2=-1$. Its integer powers repeat with period four.


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