Total Distance Traveled

Calculus Level 2

An athlete is able to jump forever. However, everytime that she jumps she gets a bit more tired, and every jump goes 1 2 \frac{1}{2} as far as her prior jump. Now, for her very first jump, she goes 1 2 \frac{1}{2} of a foot. On her second jump, she goes 1 4 \frac{1}{4} of a foot, and so on and so forth. The beginning of her journey is shown in the gif above.

What is the total distance traveled?


The answer is 1.

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Chew-Seong Cheong
Aug 18, 2018

Surprisingly only just a foot. It is a case of infinite sum of geometric progression.

S = a 1 − r where a is the first term and = 1 2 1 − 1 2 r is the common ratio. = 1 \begin{aligned} S& =\frac a{1-r} & \small \color{#3D99F6} \text{where }a \text{ is the first term and} \\ &= \frac {\frac 12}{1-\frac 12}& \small \color{#3D99F6} r \text{ is the common ratio.} \\ & = \boxed{1} \end{aligned}

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