20030702, 18:01  #1 
Oct 2002
5×7 Posts 
Another Series
When someone posts a series and asks what number comes next, in theory there are an infinite number of answers. All you need to do is to come up with a polynomial of degree n+1 where n is the number of terms in the given series. The first n solutions to the polynomial matching the first n terms in the series. You can then have another solution to the polynomial that is any value you desire.
Of course, when someone posts a series, there is the implied requirement that the rule be the simplest one possible and a high degree polynomial is not very simple. With that in mind, what digits come next? (And why?) 3.141592... 
20030702, 18:08  #2 
Nov 2002
2×37 Posts 
I think the next numbers must be
3,141592653589793238462643383279502 because they are the digits of pi!! greetz andi314 :D 
20030702, 18:24  #3  
Oct 2002
5·7 Posts 
Quote:
The simplest solution is 355/113 So the series really continues: 3.141592920353982300884955752212... 

20030702, 19:09  #4 
Jun 2003
The Texas Hill Country
3^{2}·11^{2} Posts 
Another Series
That all depends on your metric for simplicity.
Your expression is a quotient that requires 7 symbols. On many systems, I can express the other number with only one symbol. 
20030702, 22:58  #5  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
Quote:
Pi has far more fundamental significance, and appears much more often in an immense variety of contexts, than 355/113. Quote:
But that's really a lazy answer which dodges sincere effort at using one's intelligence to determine the most logical or simplest continuation within the context of the problem. Quote:
Once one has learned about fitting polynomials to given points, one can trot out this answer automatically (or for humorous intent) in response to "continue the series" problems  as my friends and I did when we were young  but that doesn't make it the most intelligent answer in most contexts. 

20030703, 01:07  #6 
Sep 2002
2×3×139 Posts 
I'm definitely out of my league if people were doing these things when they were children and they always completely elude me.

20030703, 01:18  #7 
"Richard B. Woods"
Aug 2002
Wisconsin USA
7692_{10} Posts 
Oh, don't be fooled by my photo! It's not recent! ;)
When I wrote "young" I meant "young adult". 
20030703, 08:32  #8  
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
2^{4}×719 Posts 
Quote:
Ah, by "simplest" you mean the rational fraction with the smallest denomimator. Fair enough. There is at least a simple algorithm for determining it: the continued fraction expansion. Paul 

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