Question

An athlete is able to jump FOREVER. However, every time she jumps (Long Jump) she gets a bit more tired, and every jump goes 1/2 as far as her prior jump. Now, for her very first jump, she goes 1/2 of a foot. On her second jump, she goes ¼ of a foot, and so on and so forth. How many jumps does it take for her to travel 1 foot? Show Process by using a number line.

Answer 1

Final Answer : She will never complete 1 foot

Steps:
1) We will find the maximum value of this process, when this is carried out infinite times.

This process goes like a Infinite GP, whose
First term,a = 1/2ft.
Common Ratio, r = 1/2ft.
$s = \frac{a}{1 - r} = \frac{ \frac{1}{2} }{1 - \frac{1}{2} } = \frac{ \frac{1}{2} }{ \frac{1}{2} } = 1$


Maximum value of this process is One Foot when the athlete jumps forever or Practically, she will never complete her 1 foot.

Hence, Athlete will never complete. her complete .
Here,
As shown in Number line, since we are taking half distance of the previous one.
Starting from 1/2 ft. ,and ending at 1 foot.
The value always tends towardd 1 but never reaches one.

Answer 2

So, as per the question
she starts with 1/2 foot distance, and her ability reduces by 1/2 factor for each step she takes
Now,
The distance covered can be defined in the form of geometric series as,
1/2 + 1/4+  1/8+  1/16 ...
Now,
Sum of an GP for r<1 is
S=a(1- r^n)/(1-r)
Do, she has to complete 1 foot no matter how many number of steps she takes
Also,
S∞ = a/(1-r)
Now,
a=1/2
r=1/2
So,
S∞ = 1
By this we conclude that she has to take infinite number of steps to complete a distance of 1 foot
(It was also mentioned in the question that she can run forever)
So,
if we want to represent it on number line
one end will be a point starting om 1/2 the other end will be arrow tending to ∞