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Series Question
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Michael Ward
science forum beginner


Joined: 30 Apr 2005
Posts: 22

PostPosted: Thu Jul 07, 2005 10:56 pm    Post subject: Series Question Reply with quote

Hey guys need some help on this.

You may have heard the fable about the touroise and the hare. Suppose that the tortoise and the hare are running a 5000m race. The tortoise proceeds very slowly, never changing its speed. The hare runs very quickly at the start. The tortoise travels 10 m every minute. The hare travels 2500 m in the first minute but, in each minute after that, travels only half of the remaining distance (in the second minute, the hare travels half of the ramaining 2500 m, and so on).

a) Write the series for the distance the tortoise travels for the first few minutes of the race, where Sn represents the total distance travelled after n minutes. Each term of the series will represent the distance the tortoise travels in that minute.

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 will represent the distance the hare travels in that minute.

c) Analyze and 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 for your analysis.

Heres what I got so far.

A) says take the first few mins, So do I right it like
S5 10, 10, 10, 10 ,10

B) S5 2500, 1250, 625, 312. 50, 156.25

C) Tort takes 500 minutes to travel 5000 m which is obvious.
As for the hare i'm having trouble answering I got was 1250 minutes to travel the 5000 m
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Lynn Kurtz
science forum Guru


Joined: 02 May 2005
Posts: 603

PostPosted: Thu Jul 07, 2005 10:56 pm    Post subject: Re: Series Question Reply with quote

On Thu, 07 Jul 2005 18:56:03 EDT, Max <mikeward1980@yahoo.com> wrote:

Quote:
Hey guys need some help on this.

You may have heard the fable about the touroise and the hare. Suppose that the tortoise and the hare are running a 5000m race. The tortoise proceeds very slowly, never changing its speed. The hare runs very quickly at the start. The tortoise travels 10 m every minute. The hare travels 2500 m in the first minute but, in each minute after that, travels only half of the remaining distance (in the second minute, the hare travels half of the ramaining 2500 m, and so on).

a) Write the series for the distance the tortoise travels for the first few minutes of the race, where Sn represents the total distance travelled after n minutes. Each term of the series will represent the distance the tortoise travels in that minute.

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 will represent the distance the hare travels in that minute.

c) Analyze and 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 for your analysis.

Heres what I got so far.

A) says take the first few mins, So do I right it like
S5 10, 10, 10, 10 ,10

B) S5 2500, 1250, 625, 312. 50, 156.25

C) Tort takes 500 minutes to travel 5000 m which is obvious.
As for the hare i'm having trouble answering I got was 1250 minutes to travel the 5000 m

You will never get it figured out writing decimals an in B). Write the
tortoise series in the form of a geometric series so it looks like:

a + ar + ar^2 + ar^3...ar^(n-1)

with your numbers leaving it "unsimplified". Then look in your algebra
book for how to sum n terms of a geometric series. Then see how far it
has gotten in 500 minutes.

--Lynn
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Lynn Kurtz
science forum Guru


Joined: 02 May 2005
Posts: 603

PostPosted: Thu Jul 07, 2005 10:56 pm    Post subject: Re: Series Question Reply with quote

On Thu, 07 Jul 2005 23:14:48 GMT, Lynn Kurtz
<kurtzDELETE-THIS@asu.edu> wrote:


Quote:
You will never get it figured out writing decimals an in B). Write the
tortoise series in the form of a geometric series so it looks like:

...typo alert, write the *hare* series..
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Michael Ward
science forum beginner


Joined: 30 Apr 2005
Posts: 22

PostPosted: Fri Jul 08, 2005 12:58 am    Post subject: Re: Series Question Reply with quote

Sorry, I'm lost on this. So I write
2500 5000 2500000 that cant be right.
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Paul Sperry
science forum Guru


Joined: 08 May 2005
Posts: 371

PostPosted: Fri Jul 08, 2005 1:37 am    Post subject: Re: Series Question Reply with quote

In article
<24875054.1120784335296.JavaMail.jakarta@nitrogen.mathforum.org>, Max
<mikeward1980@yahoo.com> wrote:

Quote:
Sorry, I'm lost on this. So I write
2500 5000 2500000 that cant be right.

Lynn was suggesting that you let a = 2500 - to save writing if nothing
else - then...

For Br'er Hare:

S(1) = a;
S(2) = a + (1/2)a;
S(3) = a + (1/2)a + (1/2)(1/2)a = a + (1/2)a + (1/2)^2a;
S(4) = a + (1/2)a + (1/2)^2a + (1/2)^3a;
etc.

What does S(n) look like? Look up "Geometric Series" for a formula;
plug 500 into the formula.

It is almost always a bad idea to do arithmetic until the very end of a
problem.

You will discover that the hare _never_ gets there.

--
Paul Sperry
Columbia, SC (USA)
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Anon1
science forum Guru Wannabe


Joined: 03 Jun 2005
Posts: 184

PostPosted: Fri Jul 08, 2005 1:53 am    Post subject: Re: Series Question Reply with quote

On Thu, 07 Jul 2005 20:58:25 EDT, Max <mikeward1980@yahoo.com> wrote:

Quote:
Sorry, I'm lost on this. So I write
2500 5000 2500000 that cant be right.

Correct, that can't be right. The distance accumulates by 1/2 of
5000, then (1/2 of 1/2), then (1/2 of 1/2 of 1/2), etc.

Per the post by Lynn Kurtz, consider the geometric series

a + ar + ar^2 + ar^3 + ... + ar^(n-1)

In you original post you found the hare series to be

2500 + 1250 + 625 + ... which is

2500 + 2500(1/2) + 2500(1/2)(1/2) + ...

In the geometric series, what are the values of a and r? Now, read
the Kurtz post again. 8-)

HTH
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Dana
science forum addict


Joined: 31 May 2005
Posts: 71

PostPosted: Fri Jul 08, 2005 1:21 pm    Post subject: Re: Series Question Reply with quote

Hi. If you would like, the distance of each minute could be:

625*2^(3 - n)

and the total distance given by:

625*2^(3 - n)*(2^n - 1)

--
Dana


"Max" <mikeward1980@yahoo.com> wrote in message
news:16637694.1120776993556.JavaMail.jakarta@nitrogen.mathforum.org...
Quote:
Hey guys need some help on this.

You may have heard the fable about the touroise and the hare. Suppose
that the tortoise and the hare are running a 5000m race. The tortoise
proceeds very slowly, never changing its speed. The hare runs very
quickly at the start. The tortoise travels 10 m every minute. The hare
travels 2500 m in the first minute but, in each minute after that, travels
only half of the remaining distance (in the second minute, the hare
travels half of the ramaining 2500 m, and so on).

a) Write the series for the distance the tortoise travels for the first
few minutes of the race, where Sn represents the total distance travelled
after n minutes. Each term of the series will represent the distance the
tortoise travels in that minute.

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 will represent the distance the
hare travels in that minute.

c) Analyze and 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 for your analysis.

Heres what I got so far.

A) says take the first few mins, So do I right it like
S5 10, 10, 10, 10 ,10

B) S5 2500, 1250, 625, 312. 50, 156.25

C) Tort takes 500 minutes to travel 5000 m which is obvious.
As for the hare i'm having trouble answering I got was 1250 minutes to
travel the 5000 m
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William Ashley
science forum beginner


Joined: 25 Jun 2006
Posts: 1

PostPosted: Sun Jun 25, 2006 10:50 pm    Post subject: Re: Series Question Reply with quote

Hi,

I've been looking at this question. I think one of the issues is calculation becomes very small. If I am correct in thinking the ratio (r) is 1/2 or 0.5 . If this is the case the number to a high power yeilds a very LOW number. Which makes the 2500 * x to the 0.0000000000000000000000000000000000000000000000000 etc... very very small. The calculator returns "0", this is not true but it goes beyond the "regular" floating point value for the calc I geuss. Knowing this is geometric we calculate this using the series 2500, 1250, 625 ... etc.. where t_n = (t_n-1)/2^n and S_n = A + AR + AR^2 + AR^3 +... + AR^n-1

where S_500 = AR^500-1 OR AR^499
sub A = 2500
sub R = 0.5
in AR^499 for value of S_500
S_500 = 2500(0.5^499)
0.5^499 returns error number (0) we know this is not true though. In Sci mode it returns 0E0 using a lower power numbernumber such as 0.5^250 the number is 5.527...^-76 which we know to be very small 0.0000000000000000000000000000000000000000000000000000000
0000000000000005 etc.. something like that. it is unworkable in the calc with current knowledge.
suffice to say it is an extremely large number. What we do know is that the number is div 2 every time and we are always 1/2 of the last number short. The rabit can 'never' finish, as it becomes infintismally small. (However the hare hair 'will' grow. So whatever rate the hare hair is growing at will eventually allow the rabit to finish in a day or so.
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G.E. Ivey
science forum Guru


Joined: 29 Apr 2005
Posts: 308

PostPosted: Tue Jun 27, 2006 1:47 pm    Post subject: Re: Series Question Reply with quote

Quote:
Hey guys need some help on this.

You may have heard the fable about the touroise and
the hare. Suppose that the tortoise and the hare are
running a 5000m race. The tortoise proceeds very
slowly, never changing its speed. The hare runs very
quickly at the start. The tortoise travels 10 m
every minute. The hare travels 2500 m in the first
minute but, in each minute after that, travels only
half of the remaining distance (in the second minute,
the hare travels half of the ramaining 2500 m, and so
on).

a) Write the series for the distance the tortoise
travels for the first few minutes of the race, where
Sn represents the total distance travelled after n
minutes. Each term of the series will represent the
distance the tortoise travels in that minute.

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 will represent the
distance the hare travels in that minute.

c) Analyze and 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 for your analysis.

Heres what I got so far.

A) says take the first few mins, So do I right it
like
S5 10, 10, 10, 10 ,10

This (the "easy" part!) is incorrect. Sn is supposed to be the TOTAL distance traveled. The sequence is

10, 20, 30, 40, 50, ..., an arithmetic sequence.

Quote:
B) S5 2500, 1250, 625, 312. 50, 156.25

2500, 2500+ (1/2)2500, 2500+ (1/2)2500+ (1/4)2500,

2500+ (1/2)2500+ (1/4)2500+ (1/8)2500= 2500(1+ 1/2+ 1/4+ 1/Cool, ... a geometric series.

Quote:
C) Tort takes 500 minutes to travel 5000 m which is
obvious.
As for the hare i'm having trouble answering I got
was 1250 minutes to travel the 5000 m
A heckuva lot longer than that!
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Dana DeLouis
science forum beginner


Joined: 06 Mar 2006
Posts: 37

PostPosted: Tue Jun 27, 2006 11:44 pm    Post subject: Re: Series Question Reply with quote

Quote:
b) Write the series for the distance the hare travels
for the first few minutes of the race...
...Each term of the series will represent the
distance the hare travels in that minute.

I'll cheat with Mathematica...

equ = {d[1] == 2500, d[t] == d[t - 1]/2};

RSolve[equ, d[t], t]

{d[t] -> 625*2^(3 - t)}

First few distances are:

Table[625.*2^(3 - t), {t, 4}]

{2500., 1250., 625., 312.5}

And the limit at time 't' at infinity...

Sum[625*2^(3 - t), {t, 1, Infinity}]
5000

The limit at time "infinity" is the length of the race...5,000.

--
HTH. :>)

Dana DeLouis
Windows XP, Office 2003, Mathematica 5.2



"G.E. Ivey" <george.ivey@gallaudet.edu> wrote in message
news:13397268.1151416067568.JavaMail.jakarta@nitrogen.mathforum.org...
Quote:
Hey guys need some help on this.

You may have heard the fable about the touroise and
the hare. Suppose that the tortoise and the hare are
running a 5000m race. The tortoise proceeds very
slowly, never changing its speed. The hare runs very
quickly at the start. The tortoise travels 10 m
every minute. The hare travels 2500 m in the first
minute but, in each minute after that, travels only
half of the remaining distance (in the second minute,
the hare travels half of the ramaining 2500 m, and so
on).

a) Write the series for the distance the tortoise
travels for the first few minutes of the race, where
Sn represents the total distance travelled after n
minutes. Each term of the series will represent the
distance the tortoise travels in that minute.

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 will represent the
distance the hare travels in that minute.

c) Analyze and 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 for your analysis.

Heres what I got so far.

A) says take the first few mins, So do I right it
like
S5 10, 10, 10, 10 ,10

This (the "easy" part!) is incorrect. Sn is supposed to be the TOTAL
distance traveled. The sequence is
10, 20, 30, 40, 50, ..., an arithmetic sequence.

B) S5 2500, 1250, 625, 312. 50, 156.25

2500, 2500+ (1/2)2500, 2500+ (1/2)2500+ (1/4)2500,
2500+ (1/2)2500+ (1/4)2500+ (1/8)2500= 2500(1+ 1/2+ 1/4+ 1/Cool, ... a
geometric series.

C) Tort takes 500 minutes to travel 5000 m which is
obvious.
As for the hare i'm having trouble answering I got
was 1250 minutes to travel the 5000 m
A heckuva lot longer than that!
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