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# help

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A bug of negligible size starts at the origin of the coordinate plane. First, it moves 1 unit right to (1,0). Then it makes a 90 degree turn counterclockwise and travels 1/2​ a unit to (1,1/2).  It continues in this fashion endlessly, each time making a 90 degree turn counterclockwise and traveling half as far as in the previous move.  What point does the bug end up at?

Dec 18, 2019

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A bug of negligible size starts at the origin of the coordinate plane.
First, it moves 1 unit right to (1,0).
Then it makes a 90 degree turn counterclockwise and travels 1/2? a unit to (1,1/2).
It continues in this fashion endlessly, each time making a 90 degree

turn counterclockwise and traveling half as far as in the previous move.
What point does the bug end up at?

$$\text{Let the point the bug does end up at (x,y)}$$

Geometric progression:

$$\begin{array}{|rcll|} \hline x &=& 1-\dfrac{1}{2^2}+\dfrac{1}{2^4}-\dfrac{1}{2^6}\pm\ldots \quad | \quad \text{Geometric progression:} \quad a=1,\ r=-\dfrac{1}{4} \\ x &=& \dfrac{1}{1- \left(-\dfrac{1}{4}\right) } \\ x &=& \dfrac{1}{1+ \left(\dfrac{1}{4}\right) } \\ x &=& \dfrac{1}{ \left(\dfrac{5}{4}\right) } \\ \mathbf{x} &=& \mathbf{\dfrac{4}{5}} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline y &=& \dfrac{1}{2^1}-\dfrac{1}{2^3}+\dfrac{1}{2^5}-\dfrac{1}{2^7}\pm\ldots \\ y &=& \dfrac{1}{2^1}\left(\underbrace{ 1-\dfrac{1}{2^2}+\dfrac{1}{2^4}-\dfrac{1}{2^6}\pm\ldots }_{=x}\right) \\ y &=& \dfrac{x}{2} \\\\ \mathbf{y} &=& \mathbf{\dfrac{2}{5}} \\ \hline \end{array}$$

Dec 19, 2019