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(B) The particle moves in a sequence of steps. The $x$-coordinates and $y$-coordinates change as follows:$x$-coordinates: $1, 1, 1 + \frac{1}{4}, 1 +

Vedclass · Accepted Answer

$(\frac{4}{3}, \frac{2}{5})$

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(B) The particle moves in a sequence of steps. The $x$-coordinates and $y$-coordinates change as follows:$x$-coordinates: $1, 1, 1 + \frac{1}{4}, 1 + \frac{1}{4}, 1 + \frac{1}{4} + \frac{1}{16}, \dots$This is a geometric series for the horizontal movements: $x_{\infty} = 1 + \frac{1}{4} + \frac{1}{16} + \dots = \frac{1}{1 - 1/4} = \frac{1}{3/4} = \frac{4}{3}$.$y$-coordinates: $0, \frac{1}{2}, \frac{1}{2}, \frac{1}{2} - \frac{1}{8}, \frac{1}{2} - \frac{1}{8}, \dots$This is a geometric series for the vertical movements: $y_{\infty} = \frac{1}{2} - \frac{1}{8} + \frac{1}{32} - \dots = \frac{1/2}{1 - (-1/4)} = \frac{1/2}{5/4} = \frac{2}{5}$.Thus, $(\alpha, \beta) = (\frac{4}{3}, \frac{2}{5})$.

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