SOLUTION: The fable about the tortoise an the hare? what? lol (math)? hello this is a math question that confused me from the first line. I do need help. Sorry if its a little long.

Question 1022931: The fable about the tortoise an the hare? what? lol (math)?
hello this is a math question that confused me from the first line. I do need help.
Sorry if its a little long.
You may have heard the fable about the tortoise and the hare. Suppose the tortise and he hare are running a 2000 m race. the tortoise proceeds very slow, never changing its speed.
The hare runs very quickly at the start.
The tortoise travels 20m every minute. The hare travels 1000m in the 1st minute but, in each minute, after that, it travels only half of the remaining distance. (in the 2nd minute, the hare travels half of the remaining 1000 m and so on)
a) write the series for the distance the tortoise travels for the first few minutes of each race, where Sn represents the total distance traveled after n minutes.
Each term of the series will represent the distance the tortoise travels in that minute.
(what?? so confused lol)
b) write the series for the distance the hare travels for the first few minutes of the race, where Sn represents the total distance traveled after n minutes. Each term of the series represents the distance the hare travels in that minute.
so I guess they want me to do one for the Tortoise and one for the Hare.
then finally
c) Analyze an compare the two series. include information such as Sn for both the tortoise and the hare and your conclusion as to who will win the race. provide support.
Found 2 solutions by rothauserc, robertb:
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
a) the tortoise series is an arithmetic series and
:
Sn = 20 + 40 + 60 + .....+ n = (n/2)(a1 + an) where n is the number of terms to sum and a1 is the first term and an is the nth term
:
each term, an = a1 + d(n-1), where a1 is the first term, d is the common difference, and n is the nth term
:
d = 20
********************************************************************************
b) the hare series is a geometric series and
:
Sn = 1000 + 500 + 250 + ....... + n = a((1 - r^n) / (1 - r)), where a is the first term and r is the common ratio
:
each term, an = ar^(n-1), where a is the first term and n is the nth term
:
r = (1/2)
:
*******************************************************************************
c) analysis
: for the tortoise, 2000 = (n/2)(20 + an), and tortoise crosses the finish line in 14 minutes, that is, n = 14
:
for the hare, the common ratio is (1/2) < 1
:
we know that the limit as n ----> infinity of a geometric sum is
:
a / (1 - r) = 1000 / (1/2) = 2000
:
the hare will approach the finish line but never cross it :-)
:

Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
a) +...+20+20 = 20n <--- twenty n's here
(The hare runs 20 meters at every minute.)

b) +125+...+
The sum for the first n terms of a geometric sequence is
c)The tortoise will arrive at the finish line after 100 minutes. (20n = 2000 ==> n=100 minutes.)
At n=100 minutes, the hare would have traveled meters, which is really close to 2000, but not quite the finish line yet, because there is still an infinitesimal distance remaining.

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