Sequence/ Series Tortoise and the Hare

[BeccaP]

You may have heard the fable about the tortoise and the hare. Suppose the tortoise and the hare are running a 2000 m race. The tortoise proceeds very slowly, never changing its speed. The hare runs very quickly at the start. The tortoise travels 20m every minute. The hare travels 1000 m in the 1st minute but, in each minute after that, travels only half of the remaining distance (in the 2nd minute, the hare travels half of the remaining 1000 m and so on).

B) Write the series for the distance the hare travels for the first few minutes of the race, where Sn represents the total distance travelled after n minutes.
Each term of the series represents the distance the hare travels each minute.

I found that the tortoise finishes the race in 100 minutes. As such the hare must finish the race in or before 100 minutes also or lose the race.

The sequence for the hare is:

1000, 500, 250, 125, 62.5, 31.25 ...

a= 1000
r= 0.5

n= 100?

The formula for a geometric series is:

Sn = a (1-r^n)
___________ <----(division sign sorry)
1 - r

Now I have plugged the numbers into the equation...and got the answer 2000...but it doesn't make sense because by dividing by 2 the hare will not finish the race. The number will get smaller and smaller but will not reach a definitely zero. I think? Plus it specifically asks later in the question who wins...they can't tie then?

Help would be greatly appreciated thank you!

 

[tkhunny]

and got the answer 2000...but it doesn't make sense because by dividing by 2 the hare will not finish the race.

This makes no sense. Dividing by 2 is part of the formulation. How can you blame anything on dividing by 2? If you don't divide by 2, you have the wrong result.

You have "2000" ONLY because of rounding. You can express successive sums another way.

1 term: 1000 = 2000 - 1000

2 terms: 1000 + 500 = 2000 - 1000*(1/2)

3 terms: 1000 + 500 + 250 = 2000 - 1000*(1/2)^2

...

n terms: 1000 + 500 + 250 + ... + 1000*(1/2)^(n-1) = 2000 - 1000*(1/2)^(n-1)

...

100 terms: 1000 + 500 + 250 + ... + 1000*(1/2)^(99) = 2000 - 1000*(1/2)^(99) <== Not quite 2000, is it?

Mr. Hare will not be finishing the race in any finite chunk of time.