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Thread: 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: .$\displaystyle 10$

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

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

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

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

    . . and so on . . .


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

    $\displaystyle \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 $\displaystyle a = 1$, common ratio $\displaystyle r = \tfrac{1}{2}$
    . . Its sum is: .$\displaystyle \frac{1}{1-\frac{1}{2}} \:=\:2$


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

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