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f: N -> N is a strictly increasing function such that f(f(n))=3n

What is the value of f(2001)?

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

Hi;

**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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How?

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

Hi;

**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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I know.

Further information, please?

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

Using the problem itself (RIPOSTP) we can get a bunch of values

{2, 3, 6, 7, 8, 9, 12, 15, 18, 19, 20,...} for n = 1, 2, 3...

Now we just use EM.

**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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**Olinguito****Member**- Registered: 2014-08-12
- Posts: 649

Hi Agnishom.

Let me try. We have

.As f is strictly increasing, we must have . It can’t be 1 otherwise . It can’t be 3 either otherwise so which is impossible for a strictly increasing function. Therefore: by strict-increasingness.Now note that . This gives and and if then while if then Thus and . Thus .*Last edited by Olinguito (2014-09-02 10:50:33)*

*Bassaricyon neblina*

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**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 580

Anyone able to attack this problem head on by figuring what f(n) is. No piecewise functions, just one expression?

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Olinguito wrote:

This gives

and and if then while if then

I don't get this. How do you get this?

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

Hi;

Anyone able to attack this problem head on by figuring what f(n) is.

To some extent that can be done. It leads to the following rote procedure for the solution.

We expand on Michael Somos' answer and continue it. He provides 3 recurrences,

Using these we can recursively solve the problem by redefining it in terms of easier quantities.

Cleaning it up we get:

Now remembering our sequence,

you can see that a(1) = 2 and a(2) = 3 so

Looks complicated but it is just substitutions until the whole problem is defined in terms of the first two terms. I deliberately did not simplify each expression to make it easier to see the tree.

**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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**Olinguito****Member**- Registered: 2014-08-12
- Posts: 649

Agnishom wrote:

and come from ; they can be proved by induction.Note that there are integers strictly between and , and also integers strictly between Olinguito wrote:This gives

and and if then while if thenI don't get this. How do you get this?

and

Hence, by strict-increasingness, for . Get it? Put another way:- If there are
*m*integers strictly between*a*and*b*and*m*integers strictly between f(*(a*) and f(*b*) and f is strictly increasing, then f(*a*+*r*) = f(*a*) +*r*for 1 ≤*r*≤*m*,

*Last edited by Olinguito (2014-09-02 10:50:12)*

*Bassaricyon neblina*

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bobbym wrote:

Hi;

Anyone able to attack this problem head on by figuring what f(n) is.

To some extent that can be done. It leads to the following rote procedure for the solution.

We expand on M Somos's answer and continue it. He provides 3 recurrences,

Using these we can recursively solve the problem by redefining it in terms of easier quantities.

Cleaning it up we get:

Now remembering our sequence,

you can see that a(1) = 2 and a(2) = 3 so

Looks complicated but it is just substitutions until the whole problem is defined in terms of the first two terms. I deliberately did not simplify each expression to make it easier to see the tree.

your code, please?

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

There is no code. That was done by hand with a pen. It requires no mathematics at all, just arithmetic on the first 2 initial conditions. M Somos made it easy but EVW was also able to derive his recurrences.

**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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**Olinguito****Member**- Registered: 2014-08-12
- Posts: 649

I didn’t realize this was supposed to be a computer question and solved it the hard way.

*Bassaricyon neblina*

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

This is not a computer answer. It was done by pen and paper in the manner shown. The only requirement is arithmetic. See post #13

**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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Olinguito wrote:

I didn’t realize this was supposed to be a computer question and solved it the hard way.

I like your solution very much except that a solution is simpler when you can program it. (opinion 37)

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

It took 5 minutes and 30 minutes to latex it by hand.

Regarding Opinion #37:

DZ wrote:

You would realize that you REALLY understand something ONLY after you PROGRAMED it YOURSELF.

But as I have said repeatedly my working here is not computer related.

**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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M has an option to copy latex

I do not understand integer addition. I cannot program it.

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

I am using M's much vaunted latex capabilities. I have even modified them. But latex as he sees it and latex on the forum are not the same things.

I do not understand integer addition. I cannot program it.

You do not understand it, you just think you do.

**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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All my programs are based on addition, so I understand nothing?

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

It is possible to use words and concepts that you do not fully understand, effectively. If humanity were required to have perfect understanding of things we would not exist. God would have made something better. Obviously perfect understanding or even a good understanding is not necessary for survival in math or life. See my signature.

**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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There are people who can program integer addition. E.g, bobbym

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

I can program integer addition even multiprecision on very primitive machines but I do not understand many things about it. I think humans get caught up in a very simple trap. Let us say that person A is a lot smarter than most of the other humans. Soon A becomes pride filled. A forgets that just because he/she is superior to other humans does not imply great intelligence or abilities.

**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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What is J-rod?

The person below is C

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

We are getting way off the topic. Were you able to follow my solution?

**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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