Another Fibonacci-related race? Seriously?

Calculus Level 5

The city of Mathville is once again having a race. This time, however, competitiors will start at the city centre and will run 1km east, 1km north, \(\dfrac{1}{2}\)km east, \(\dfrac{1}{3}\)km north, \(\dfrac{1}{5}\)km east\(\dfrac{1}{8}\)km north ad infinitum.

If the distance displaced by the competitors can be represented as \(k\) metres, what is \(\lfloor{k}\rfloor\)?

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