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#1 2006-02-27 02:20:06

Jai Ganesh
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Registered: 2005-06-28
Posts: 48,111

Series and Progressions

SP # 1

If p, q, r are in Arithmetic Progression and x, y, z are in Geometric Progression, show that


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#2 2006-02-27 03:00:43

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Series and Progressions

Let:
p=p
q=p+a
r=p+2a
x=x
y=bx
z=b^2x
Then:

Last edited by krassi_holmz (2006-02-27 03:05:26)


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#3 2006-02-27 16:41:27

Jai Ganesh
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Posts: 48,111

Re: Series and Progressions

krassi_holmz, although I don't see any serious mistake in the way you started, I am not fully convinced with the proof. I shall wait for a few more days before posing the solution.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#4 2006-02-27 17:38:34

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Series and Progressions

p,q and r are in Arithmetic prgression, so
q=p+a
r=p+2a, because of the arithmetic progression propeties.
Same for the x,y,z:
y=bx
z=b^2x

Next is just simple arithmetic reduction:



Where's my mistake?

Last edited by krassi_holmz (2006-02-27 17:46:07)


IPBLE:  Increasing Performance By Lowering Expectations.

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#5 2006-02-27 18:05:48

Jai Ganesh
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Registered: 2005-06-28
Posts: 48,111

Re: Series and Progressions

correct_answer.gif


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#6 2006-02-27 18:16:13

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Series and Progressions

That's better.
smile


IPBLE:  Increasing Performance By Lowering Expectations.

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#7 2006-02-28 16:14:27

Jai Ganesh
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Registered: 2005-06-28
Posts: 48,111

Re: Series and Progressions

SP # 2

The sum of the digits of a three digit number is 12. The digits are in Arithmetic Progression. If the digits are reversed, then the number is diminished by 396. Find the number.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#8 2006-02-28 17:56:26

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Series and Progressions

642?


IPBLE:  Increasing Performance By Lowering Expectations.

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#9 2006-02-28 19:43:27

Jai Ganesh
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Registered: 2005-06-28
Posts: 48,111

Re: Series and Progressions

correct.gif


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#10 2006-02-28 19:45:45

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Series and Progressions

I want MORE!!!


IPBLE:  Increasing Performance By Lowering Expectations.

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#11 2006-03-01 18:02:41

Jai Ganesh
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Posts: 48,111

Re: Series and Progressions

Here you get!

SP# 3

The sum of an infinite series in Geometric Progression is 57 and sum of their cubes is 9747. Find the series.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#12 2006-03-02 17:43:42

Jai Ganesh
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Registered: 2005-06-28
Posts: 48,111

Re: Series and Progressions

SP # 4

A ball is dropped from a height of 6m and on each bounce it rebounces to 2/3 of its previous height. How far does the ball travel till it stops bouncing?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#13 2006-03-02 17:52:01

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Series and Progressions

SP #4: the ball is dropped, so it doesn't travel anywhere.

But seriously, by traveled, do you mean both positive and negative changes in height?  In other words, do we count the ball going up and down?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#14 2006-03-03 00:11:41

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Series and Progressions

If we count this we get the sum :


I may be wrong.


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#15 2006-03-03 02:27:52

Jai Ganesh
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Registered: 2005-06-28
Posts: 48,111

Re: Series and Progressions

excellent.gif


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#16 2006-03-03 02:36:28

Jai Ganesh
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Registered: 2005-06-28
Posts: 48,111

Re: Series and Progressions

This is how the problem is solved in a different way.

1. The distance covered in the downward path is an infinite Geometric series with a=6m, r=2/3.
Therefore, S[sub]n=[6/(1-2/3)]=6/(1/3)=18m
2. The distance covered in the upward path is an infinte Geometric series with a=4m, r=2/3.
S[sub]n=[4/(1-2/3)]=4/(1/3)=12m

Total distance = 18m + 12m = 30m.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#17 2006-03-03 23:33:10

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: Series and Progressions

Ricky wrote:

SP #4: the ball is dropped, so it doesn't travel anywhere.

If you're being picky like that, then technically it travels 6m. tongue


Why did the vector cross the road?
It wanted to be normal.

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#18 2006-03-04 00:47:40

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

Re: Series and Progressions

Einstein would say:
It depends on it's speed.
tongue tongue


IPBLE:  Increasing Performance By Lowering Expectations.

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#19 2006-03-05 16:49:00

Jai Ganesh
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Registered: 2005-06-28
Posts: 48,111

Re: Series and Progressions

SP # 5

The first term of a Geometric Progression is 64 and the average of the first and the fourth terms is 140. Find the common ratio 'r'.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#20 2006-03-05 17:05:52

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Series and Progressions

Last edited by Ricky (2006-03-05 17:06:02)


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#21 2006-03-05 17:20:13

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 48,111

Re: Series and Progressions

Well done. Ricky! cool


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#22 2006-03-06 18:29:26

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 48,111

Re: Series and Progressions

SP # 6

A man borrows $5,115 to be repaid in 10 monthly instalments. If each instalment is double the value of the last, find the value of the first and the last instalment.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#23 2006-03-07 04:19:08

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: Series and Progressions


Why did the vector cross the road?
It wanted to be normal.

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#24 2006-03-07 04:22:56

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 48,111

Re: Series and Progressions

You are correct, mathsyperson! Well done! cool


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#25 2006-03-07 19:38:54

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

Re: Series and Progressions

SP#3:



q=57(1-a)
Solving

a=2/3 or a=3/2;
Then q=19 or q=-57/2
But when q=-57/2 the sum is negative, so:
So the answer is:
a=2/3;q=19


IPBLE:  Increasing Performance By Lowering Expectations.

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