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#1 2015-04-30 03:26:08

Agnishom
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From: Riemann Sphere
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21 Questions from Amrita

1. If a and b are the roots of the equation

x^2 - 2 x cos(t) + 1 = 0

Prove that the equation whose roots are a^n, b^n is

x^2 - 2 x cos(n t) + 1 = 0

(Possible Hint from Agnishom: This has got something to do with complex numbers)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#2 2015-04-30 23:39:50

Bob
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Re: 21 Questions from Amrita

hi Agnishom,

Some strange things happened when I tried this and I almost rejected my method.  But I followed it through anyway, and surprise surprise, it came out as required.  So maybe this is right afterall. dizzy

So

Now if a^n and b^n are the roots of a quadratic then

Using the complex formula:

and

Using sum and product for this quadratic gives

yikes

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2015-05-01 00:16:10

Agnishom
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Re: 21 Questions from Amrita

That is ingenious. How did you figure all this out?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#4 2015-05-01 00:40:51

Bob
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Re: 21 Questions from Amrita

How indeed?  Good question.

I thought it would have to involve the sum and product formulas but got stuck with a^n + b^n

For a while I wondered about expanding (a+b)^n but that looked horrid.

So I turned to the Euler formula and realised I'd have to get an expression for sin(t).  As that was going to involve a square root I stopped without doing that; hoping something else would turn up.  When it didn't, I thought I'll have to try it.  Writing 4 = 4ab was the key.  Once I did that I got a better expression for sin(t) but the 'i' was worrying.

Even more so when it cancelled out of the e^it equation.  I thought I must have made a mistake and re-worked it several times.  When it kept coming out the same I thought I'd just have to plough on and see what would happen. So I was very surprised when it all fell into place.  It was like climbing up a difficult mountain side, getting to the top, and discovering a land of milk and honey on the other side.  Wow, where did that come from?  I'm still waiting to be told there's an error in there somewhere.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#5 2015-05-01 02:36:42

Agnishom
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Re: 21 Questions from Amrita

Great.

Originally, the question did not involve the statement which we are supposed to prove. It just asked to deduce the quadratic equation.

I did deduce it with some experimentation but could not prove it.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#6 2015-05-01 02:39:52

Agnishom
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From: Riemann Sphere
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Re: 21 Questions from Amrita

2. If

225 x^2 + 25 y^2 + 9 z^2 - 75 x y + 15 y z + 45 x z = 0

Prove that x, y, z are in an Arithmetic Progression

Last edited by Agnishom (2015-05-01 02:55:58)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#7 2015-05-01 05:26:11

bobbym
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Re: 21 Questions from Amrita

225 x^2 + 25 y^2 + 9 z^2 - 75 x *y + 15 y *z + 45 x *z = 0

Have you copied the problem correctly?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#8 2015-05-01 11:27:54

Agnishom
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Re: 21 Questions from Amrita

No, I modified the expression a bit.

Can you find counterexamples?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#9 2015-05-01 13:11:22

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: 21 Questions from Amrita

Yes I can and you should be able too.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#10 2015-05-01 20:16:07

Bob
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Re: 21 Questions from Amrita

hi

I couldn't make progress so I started from the other 'end'.  ie.  If x, x+d, x + 2d are in AP

I substituted for y and z and after some simplification got:

Starting with such an AP does gives values that fit the equation.  That doesn't constitute a proof of course but it will tell you when you'll get  an AP and when you won't.

Agnishom wrote:

No, I modified the expression a bit.

Please post the original question.

EDIT:  Wait a mo.  I've got two minuses that should be pluses.  That may make a big difference.  fresh attempt ...................................watch this space

ANOTHER EDIT:  Revised quadratic has no real solutions.

Bob

Last edited by Bob (2015-05-01 21:45:08)


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#11 2015-05-02 12:44:35

Agnishom
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Re: 21 Questions from Amrita

AmritaR wrote:

If

25(9x^2 + y^2) + 9z^2 - 15(5xy + yz + 3xz) = 0

Prove that x,y,z form an AP


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#12 2015-05-02 13:17:16

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: 21 Questions from Amrita

That is a horse of a different color.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#13 2015-05-02 13:57:36

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: 21 Questions from Amrita

Your favorite color is green.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#14 2015-05-05 01:18:25

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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Re: 21 Questions from Amrita

Help me!


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#15 2015-05-05 01:46:29

Bob
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Posts: 10,059

Re: 21 Questions from Amrita

The question in post 11 is the one I thought I was solving.  Sequences of the form x, 3x, 5x, .... fit the equation.  I haven't yet found a way to start with the equation and get to the AP.  I tried expanding (15x + 5y + 3z)^2 and got to an interesting result but still couldn't get anywhere.

Still trying ............

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#16 2015-05-05 03:06:26

Agnishom
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From: Riemann Sphere
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Posts: 24,974
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Re: 21 Questions from Amrita

There is a difference between post 6 and 11


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#17 2015-05-05 03:39:26

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: 21 Questions from Amrita

Factor the RHS over the imaginary numbers.

Set both factors equal to 0 and leave out the 1 / 4.

Eliminate the z variable.

Plug that into f(x,y,z) and factoring we get,

Set those two factors equal to 0 and solve for z.

So we have

If a sequence is in AP then the first differences must be equal.

We are done.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#18 2015-05-05 04:18:12

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: 21 Questions from Amrita

I don't get the factorization


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#19 2015-05-05 04:19:28

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: 21 Questions from Amrita

Whenever we see something we do not understand the best thing is to ask the greatest mathematician in the world whether it agrees.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#20 2015-05-05 04:23:55

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: 21 Questions from Amrita

What if it does?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#21 2015-05-05 04:26:12

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: 21 Questions from Amrita

That is a CAS assisted proof that EVW showed me. Do you know of the story of the mathematician and the two factors?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#22 2015-05-05 04:46:37

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: 21 Questions from Amrita

No, please tell me


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#23 2015-05-05 04:57:27

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: 21 Questions from Amrita

In the 1800's the great genius Lucas had proved that M67 had factors and thus could not be a prime. Frank Cole , an mathematician gave a lecture for the AMM. On one blackboard he calculated M67 = 147,573,952,589,676,412,927 and on another the factors 193,707,721 × 761,838,257,287 which he then multiplied by hand on the blacboard and which equaled 147,573,952,589,676,412,927. He then calmly walked off the podium and sat down, the only lecture where not a single word was spoken. The math types in the audience stood up and gave him a standing ovation. He did not tell how he arrived at the factors.

It is not so important how one derives an answer, that is for schoolkids. Sometimes it is intuition, sometimes a guess, sometimes a fairy queen whispering in your ear. What is important is that you prove your answer right. If you do not believe the factorization then multiply it out by hand or use M, or preferably both.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#24 2015-05-05 05:00:09

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: 21 Questions from Amrita

Multiplying that out by hand is Kaboobly Doo.

So, you are not going to tell me how to factor that?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

#25 2015-05-05 05:01:25

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: 21 Questions from Amrita

I would say by hand it would require Alan Turing or Johhny. I did it using M.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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