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

Calculus Level 4

A stair case has the property such that it travels in the pattern right, up, right, up, right, up. . . ad infinitum.

It travels $$\cos^1\left(\frac{\pi}{10}\right)$$ to the right, $$\cos^2\left(\frac{\pi}{10}\right)$$ up, $$\cos^3\left(\frac{\pi}{10}\right)$$ right, $$\cos^4\left(\frac{\pi}{10}\right)$$ up. . .

Find the total length of displacement from the beginning of this staircase to the end.

If the answer can be represented in the form $$\sqrt{a+b\sqrt{c}}$$. Find $$a+b+c$$

Inspired by my good friend Brian Charlesworth

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