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- Jan 26, 2012

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**Problem**: Starting with the vertices $P_1(0,1)$, $P_2(1,1)$, $P_3(1,0)$, $P_4(0,0)$ on a square, we construct further points as shown in the figure below: $P_5$ is the midpoint of $P_1P_2$, $P_6$ is the midpoint of $P_2P_3$, $P_7$ is the midpoint of $P_3P_4$, and so on. The polygonal spiral path $P_1P_2P_3P_4P_5P_6P_7\ldots$ approaches a point $P$ inside the square.

a) If the coordinates of $P_n$ are $(x_n,y_n)$, show that $\frac{1}{2}x_n + x_{n+1} + x_{n+2} + x_{n+3} = 2$ and find a similar equation for the $y$-coordinates.

b) Find the coordinates of $P$.

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