# Run Cody Run!

Geometry Level 3

Cody has started running in a well organized manner. He runs $$100 \text{ m}$$ east, then turns left and runs another $$10 \text{ m}$$ north, turns left and runs $$1 \text{ m},$$ again turns left and runs $$0.1 \text{ m},$$ and on next turn $$0.01 \text{ m}$$ and so on. Assuming that Cody can run in this pattern infinitely, then the displacement from his initial position can be written as $$\frac{a}{\sqrt{b}}$$ with $$a$$ and $$b$$ as positive integers and $$b$$ square-free.

What is the value of $$a \times b?$$

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