Run Cody Run!

Geometry Level 3

Cody has started running in a well-organized manner. He runs 100 m100 \text{ m} east, then turns left and runs another 10 m10 \text{ m} north, turns left and runs 1 m,1 \text{ m}, again turns left and runs 0.1 m,0.1 \text{ m}, and on the next turn 0.01 m,0.01 \text{ m}, and so on.

Assuming that Cody can run in this pattern infinitely, the displacement from his initial position can be written as ab\frac{a}{\sqrt{b}} with aa and bb being positive integers and bb square-free.

What is the value of a×b? a \times b?

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