Trevor rides Jake piggyback III

Calculus Level 5

It's a beautiful Monday morning. The wind is violent but relaxing, the sun warm and fellow, and the Trevor is riding Jake piggyback upon the viridian turf of the golf course.

Starting at \((0,0)\), Trevor tells his serf to advance by \(\dfrac{L_{n}}{\sqrt{5}^{n}}\) meters and turn \(90^{\circ}\) clockwise at the nth step, where \(L_{1} = 1\), \(L_{2} = 3\), and \(L_{n+2} = L_{n+1}+L_{n}\).

What is the magnitude of the displacement in meters Trevor and Jake have travelled from the origin? If it is of the form \(\dfrac{a}{\sqrt[b]{c}}\) where \(a\) and \(b \geq c > 1\) are positive integers, find \(a+b+c\).

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