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Math Help - Arithmetic & Geometric Progression

  1. #1
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    Arithmetic & Geometric Progression

    I need help to solve this question.

    An ant of negligible size walks a distance of 10 units in the x-y plane along the x-axis. It then turns left and goes up 5 units from its current point. If the ant continues turning left and going half the distance it had previously walked, repeating the pattern, find the co-ordinates of the point where the ant will eventually end up. Answer: ( 8, 4 )
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by puggie View Post
    I need help to solve this question.

    An ant of negligible size walks a distance of 10 units in the x-y plane along the x-axis. It then turns left and goes up 5 units from its current point. If the ant continues turning left and going half the distance it had previously walked, repeating the pattern, find the co-ordinates of the point where the ant will eventually end up. Answer: ( 8, 4 )
    Hi puggie.

    Consider its horizontal and vertical perambulations separately. Horizontally, its displacements are

    10\,-\,10\left(\frac14\right)\,+\,10\left(\frac14\right  )^2\,-\,10\left(\frac14\right)^3\,+\,\cdots

    That is, 10 units to the right, 10\left(\frac14\right) units to the left, 10\left(\frac14\right)^2 units to the right, 10\left(\frac14\right)^3 to the left, and so on.

    Vertically, it moves

    5\,-\,5\left(\frac14\right)\,+\,5\left(\frac14\right)^  2\,-\,5\left(\frac14\right)^3\,+\,\cdots

    That is, 5 units up, 5\left(\frac14\right) units down, 5\left(\frac14\right)^2 units up, 5\left(\frac14\right)^3 units down, and so on.

    Now sum these two infinite geometric series.
    Last edited by TheAbstractionist; May 20th 2009 at 05:23 AM.
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  3. #3
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    Geometric Series

    A geometric series has first term 2 with common ratio as -1/2(x + 1). When x becomes 1/3, find the sum of all odd-numbered terms of the series. Answer given as 18/5 - not sure that it is correct! Thanks
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