Why is it stairway to heaven? Can't there be an elevator?

The net distance traveled to the right is

$X = \displaystyle\sum_{k=1}^{\infty} (\cos(\frac{\pi}{10}))^{2k - 1} = \dfrac{\cos(\frac{\pi}{10})}{1 - \cos^{2}(\frac{\pi}{10})} = \dfrac{\cos(\frac{\pi}{10})}{\sin^{2}(\frac{\pi}{10})}.$

The net distance traveled upward is

$Y = \displaystyle\sum_{k=1}^{\infty} (\cos(\frac{\pi}{10}))^{2n} = \dfrac{\cos^{2}(\frac{\pi}{10})}{1 - \cos^{2}(\frac{\pi}{10})} = \dfrac{\cos^{2}(\frac{\pi}{10})}{\sin^{2}(\frac{\pi}{10})}.$

The magnitude of the displacement is then

$\sqrt{X^{2} + Y^{2}} = \dfrac{\sqrt{\cos^{2}(\frac{\pi}{10}) + \cos^{4}(\frac{\pi}{10})}}{\sin^{2}(\frac{\pi}{10})} = \dfrac{\cos(\frac{\pi}{10})\sqrt{1 + \cos^{2}(\frac{\pi}{10})}}{\sin^{2}(\frac{\pi}{10})}.$

Now $\cos(\frac{\pi}{10}) = \dfrac{\sqrt{10 + 2\sqrt{5}}}{4}$ and $\sin(\frac{\pi}{10}) = \dfrac{\sqrt{5} - 1}{4},$ so

$\sqrt{X^{2} + Y^{2}} = \dfrac{\sqrt{10 + 2\sqrt{5}}\sqrt{26 + 2\sqrt{5}}}{6 - 2\sqrt{5}} = \dfrac{\sqrt{280 + 72\sqrt{5}}}{6 - 2\sqrt{5}} =$

$\dfrac{\sqrt{70 + 18\sqrt{5}}}{3 - \sqrt{5}} \times \dfrac{\sqrt{14 + 6\sqrt{5}}}{3 + \sqrt{5}} = \dfrac{\sqrt{1520 + 672\sqrt{5}}}{4} = \sqrt{95 + 42\sqrt{5}}.$

Thus $a + b + c = 95 + 42 + 5 = \boxed{142}.$