Trevor rides Jake piggyback II

Calculus Level 4

It's a beautiful Sunday morning. The wind is calm but fills one with energy, the sun warm and yellow, and the Trevor is riding Jake piggyback upon the lush green turf of the golf course.

Starting at \((0,0)\), Trevor tells his serf to advance by \(\dfrac{n}{2^{n}}\) meters and turn \(90^{\circ}\) clockwise at the nth step.

What is the distance Trevor and Jake have travelled from the origin? If your answer is of the form \(\frac{a}{b}\) where \(a\) and \(b\) are positive coprime integers, find \(a+b\).

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