Pattern puzzle

[DivideByZero]

kudos for anyone who solves this.

 

This pattern goes on forever:

0, 1, 1/2, 2/3...

 

What would be the limit as the pattern approaches infinity. In other words, whats the last number in the infinitely long pattern?

 

example: 1.9, 1.99, 1.999 evenly approaches 2.

[doG]

kudos for anyone who solves this.

 

This pattern goes on forever:

0, 1, 1/2, 2/3...

 

What would be the limit as the pattern approaches infinity. In other words, whats the last number in the infinitely long pattern?

 

It would approach 1 as a limit...

[Stan]

I am guessing that the pattern is to use the denominator from the previous fraction as the numerator of the next fraction, then make the next denominator the same plus 1. So,

 

1, 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10....

 

If that is the pattern, then the number approaches infinitely close to 1 without ever reaching it, as DivideByZero indicated.

[thedarkshade]

kudos for anyone who solves this.

 

This pattern goes on forever:

0, 1, 1/2, 2/3...

 

What would be the limit as the pattern approaches infinity. In other words, whats the last number in the infinitely long pattern?

 

example: 1.9, 1.99, 1.999 evenly approaches 2.

If we want to find numbers between 0 and 1, then the list of number is infinite, but it never is 1. A general formula is:

[math]a < \frac{a+b}{2} < b [/math]

 

Suppose we want to find numbers between 0 and 1, so we start like this:

 

[math]0 <\frac{0+1}{2} < 1[/math] so we get

[math]0 < \frac {1}{2} < 1[/math] .. to find other numbers, we just take the middle part and do the operations according to that formula I posted above. Finding the next number would go like this:

 

[math] \frac{1}{2} < \frac{(\frac{1}{2}+1)}{2} < 1 [/math] so then we get

 

[math] \frac{1}{2} < \frac{3}{4} < 1[/math] .. then to find the next number you take the middle part of this last one and so on..

 

You can find infinite numbers between any two numbers you want;)!

[Country Boy]

kudos for anyone who solves this.

 

This pattern goes on forever:

0, 1, 1/2, 2/3...

 

What would be the limit as the pattern approaches infinity. In other words, whats the last number in the infinitely long pattern?

 

example: 1.9, 1.99, 1.999 evenly approaches 2.

As pointed out, the limit is obviously 1. I really hate the phrasing "whats the last number in the infinitely long pattern?" Obvously an "infinitely long" sequence doesn't HAVE a "last number" and it is very misleading to refer to the limit of a sequence that way.

[thedarkshade]

Obvously an "infinitely long" ...

Yes, there are infinite numbers between any two numbers, there is no number that can be the last number near 1;) !

[DivideByZero]

no limit is not 1. This is a tricky one. Look closely, you over looked a number!

[thedarkshade]

no limit is not 1. This is a tricky one. Look closely, you over looked a number!

There is no limit, there are infinite number between 0 and 1!

[DivideByZero]

no. This pattern does have a limit. And there is only one answer.

[iNow]

You need to look up the concept of a real number.

 

http://en.wikipedia.org/wiki/Real_number

 

 

That, or this is a semantic thing where the answer is [math]\frac{1}{\infty}[/math] ...

[DivideByZero]

You need to look up the concept of a real number.

 

http://en.wikipedia.org/wiki/Real_number

 

 

That, or this is a semantic thing where the answer is [math]\frac{1}{\infty}[/math] ...

 

the answer is a real number. and it is not [math]\frac{1}{\infty}[/math]

 

tell me if you need a hint

[iNow]

My nose is a bit stuffy and clogged right now... what, with allergies and all... but I do believe I smell a wild goose chase.

[DivideByZero]

Let me just give you all a broad hint.

the pattern is not [MATH]1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10[/MATH]

[timo]

The few numbers you gave clearly do not define a unique pattern. My guess is an irrational number ~1.62, but that's with guessing a particular pattern and not showing it to actually being the limit (only a fix-point), only.

[DivideByZero]

The few numbers I gave are just enough to find only one solution. If I gave one less number there would be multiple answers. If I gave you one more number the pattern would be almost obvious. Let me tell you one thing now, the limit is a very famous number. Atheist is right about it being irrational. Hope I didn't give too much away.

[timo]

In that case I'm pretty sure I'm right about the value, too. Didn't want to explicitely write it down, because the main challenge seems to be guessing the pattern you had in mind, and it would have become pretty obvious if I had given the exact value. Would be interesting to see the full proof, though. In addition to some other potential problem I see, I am not sure that the existance of a fix-point is sufficient.

EDIT: No, the number wasn't correct; I got a sign-error in the pq-formula. Should be ~0.62.

[DivideByZero]

Yup. Atheist, I think you know the answer. I still don't know why it works...

[timo]

I do . Took some time to get the mathematical proof of convergence working, though.

[doG]

The few numbers I gave are just enough to find only one solution. If I gave one less number there would be multiple answers. If I gave you one more number the pattern would be almost obvious.

 

I do agree you would likely have given it away with the next ratio but as posed, the OP was not enough to define the sequence as unique either. Perhaps that's not too revealing. I do seem to remember Donald Duck getting a lesson on it if that helps anyone.

[DivideByZero]

ok well heres the next number for those who are stuck:

0, 1, 1/2, 2/3, 3/5...