# The odd theoretical ant

Algebra Level 4

An ant travels along a straight path with a distance of $$x$$ meters. From then on, it turns left and covers $$\frac{2}{3}$$ of the straight distance it traveled before turning, and continues doing this until it eventually reaches an unknown point $$P$$ (referred to as the red point in the middle of the picture). Refer to the figure above.

The distance of $$P$$ from the ant's starting point is $$kx$$ meters, where $$k$$ is a positive constant which can be expressed in the form $$\frac{a \sqrt {b}}{c}$$, where $$\gcd(a,c) = 1$$ and $$b$$ is square free. Determine the value of $$a + b + c$$.

Challenge: Determine an explicit formula for $$k$$.

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