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Math Help - Geometric Series

  1. #1
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    Geometric Series

    A rubber ball is dropped from height 10 meters. If it rebounds aprox 1/2 the distance after each fall, use geometric series to aprox the total distance that the ball travels before coming to rest.
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  2. #2
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    Hello, ggcoralysp!

    A rubber ball is dropped from height 10 meters.
    If it rebounds aprox 1/2 the distance after each fall, use geometric series
    to approximate the total distance that the ball travels before coming to rest.
    Let's baby-talk our way through this . . .

    The ball falls 10 meters: . 10

    It bounces up 5 meters and falls 5 meters: . 2(5)

    It bounces up \tfrac{5}{2} meters and falls \tfrac{5}{2} meters: . 2\left(\tfrac{5}{2}\right)

    It bounces up \tfrac{5}{2} meters and falls \tfrac{5}{2} meters: . 2\left(\tfrac{5}{2}\right)

    It bounces up \tfrac{5}{4} meters and falls \tfrac{5}{4} meters: . 2\left(\tfrac{5}{4}\right)

    . . and so on . . .


    The total distance is: . D \;=\;10 + 2(5) + 2\left(\tfrac{5}{2}\right) + 2\left(\tfrac{5}{4}\right) + 2\left(\tfrac{5}{8}\right) + \hdots

    \text{We have: }\;D \;=\;10 + 10\underbrace{\bigg[1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \hdots\bigg]}_{\text{geometric series}} .[1]

    The geometric series has: .first term a = 1, common ratio r = \tfrac{1}{2}
    . . Its sum is: . \frac{1}{1-\frac{1}{2}} \:=\:2


    Substitute into [1]: . D \;=\;10 + 10(2) \;=\;30 meters.

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