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#51 2007-04-27 19:25:57

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

#14

Prove that if n is an odd positive integer, there exists a sequence of n consecutive integers whose sum is equal to n[sup]2[/sup].
­


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#52 2007-04-28 02:47:02

Stanley_Marsh
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Re: Jane’s exercises

#13 I surrender....


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#53 2007-06-01 02:14:54

JaneFairfax
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Registered: 2007-02-23
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Re: Jane’s exercises

#15 This is an easy one.

Prove that any even power of any odd integer leaves a remainder of 1 when divided by 4.

Last edited by JaneFairfax (2007-06-01 02:15:24)


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#54 2007-06-01 07:39:14

Stanley_Marsh
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Registered: 2006-12-13
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Re: Jane’s exercises

This one is easier

Last edited by Stanley_Marsh (2007-06-01 07:39:28)


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#55 2007-06-01 08:24:51

JaneFairfax
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Re: Jane’s exercises


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#56 2007-06-01 20:01:07

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Re: Jane’s exercises

Yah , it comes from mutiplication law of mod


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#57 2007-06-23 02:18:53

JaneFairfax
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Re: Jane’s exercises

#16


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#58 2007-08-04 01:22:16

JaneFairfax
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Re: Jane’s exercises

#17 Here is another exercise I made up myself. big_smile


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#59 2007-08-07 09:54:03

JaneFairfax
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Registered: 2007-02-23
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Re: Jane’s exercises

#18

Last edited by JaneFairfax (2007-08-07 10:48:00)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#60 2007-08-10 22:14:03

JaneFairfax
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Registered: 2007-02-23
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Re: Jane’s exercises

This result is important because I used it in proving Tony’s question: http://www.mathsisfun.com/forum/viewtopic.php?id=7807

Last edited by JaneFairfax (2007-08-11 03:10:04)


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#61 2007-08-16 20:38:09

JaneFairfax
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Re: Jane’s exercises

#19


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#62 2007-09-13 11:24:41

JaneFairfax
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Registered: 2007-02-23
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Re: Jane’s exercises

#20

I just made this problem up myself; hopefully I’ve got the maths correct. tongue

Last edited by JaneFairfax (2007-09-13 11:26:02)


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#63 2007-10-26 22:28:52

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

#21

Consider the sequence

2, 5, 9, 12, 16, 19 …

The first term is 2 and successive terms are formed by alternately adding 3 and 4 to previous terms:

Prove that


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#64 2008-02-22 03:44:32

Kurre
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Registered: 2006-07-18
Posts: 280

Re: Jane’s exercises

a more powerful result of 15:


which is two consecutive even integers, thus one can be written on the form 4k, and one on the form 4k+2, and thus k^{2n}-1 is divisible by 8, ie k^{2n} gives rest 1 when divided by 8.

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#65 2009-04-14 00:08:05

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 84,730

Re: Jane’s exercises

Hi Jane;

Answer for #21

Don't suppose anyone will ever see this but here goes:

The recurrence formula for your set of numbers is.

this has a characteristic equation of

x^3-x^2-x+1=0 which has roots of {-1,1,1}

This then is the general form of the solution:

we need to determine c1,c2, and c3 from the initial conditions

So the general solution is:

This agrees with your general solution at the bottom of problem 21


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#66 2009-04-14 01:15:29

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

bobbym. looks like you’ve got the answer. Well done! up
exlnwrk121.jpg

My solution:

Let


Then

and
are AP’s with common difference 7 and first terms 2 and 5 respectively. Thus


Note that

can be written as:

The result follows.

Last edited by JaneFairfax (2009-04-14 01:22:51)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#67 2009-04-14 01:30:59

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 84,730

Re: Jane’s exercises

Hi Jane;

   Thanks for showing me yours, I am not that happy with mine, what justification do I have for the first step? I haven't proven that a(n+3)=a(n+2)+a(n+1)-a(n) is the recursion for that sequence. Can you provide some rigor to my arguments?

bobbym


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#68 2009-04-14 04:36:48

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

Well, I’m not that familiar with techniques on solving difference equations, but your solution looked right to me so I gave you full credit for it. big_smile

I’m pretty sure your first step was correct. cool


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#69 2009-04-14 11:33:25

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 84,730

Re: Jane’s exercises

Hi Jane;

   You say you are not  very familiar with methods of solving difference equations but yet you came up with the idea of splitting the sequence into 2 coupled difference equations, each handling every other term. I salute you.


bobbym

Last edited by bobbym (2009-04-14 11:48:04)


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#70 2009-04-14 12:21:54

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

Aw, thanks. kiss Well, I thought up that problem myself, so for this particular problem I happen to have have my own solution handy. wink

At least, I discovered the following handy lemma myself:

If a sequence A has properties that alternate between its odd and even terms, let B be the subsequence of its odd-numbered terms and C the subsequence of its even-numbered terms. Find formulas for B and C separately. Then combine the formulas by rewriting A in the manner described above.

I’d love to call this “Jane’s lemma” if no-one else has discovered it before. tongue


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#71 2009-04-15 01:13:54

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,730

Re: Jane’s exercises

Hi Jane;

   I discovered that about 10 years ago but didn't notice your method of combining them and promptly forgot about the whole idea. So I guess its yours and I will always remember it as Jane's lemma.

bobbym

Last edited by bobbym (2009-04-19 18:24:25)


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

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