I discovered a formula for the nth term of any sequence of numbers

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Discussion Overview

The discussion revolves around the discovery of a formula for the nth term of any finite sequence of numbers. Participants explore the validity and applicability of this formula, questioning its generalizability beyond specific types of sequences such as arithmetic progressions. The conversation includes theoretical considerations, examples, and concerns about publication and originality.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant claims to have discovered a formula for the nth term of any finite sequence, providing an example for two numbers.
  • Another participant argues that the provided formula only applies to arithmetic progressions and is well-known, suggesting it is trivial.
  • Some participants assert that no formula can predict the next term of a general sequence based on previous terms, citing the possibility of arbitrary choices for the next term.
  • There is a suggestion that polynomial interpolation may relate to the proposed formula, with references to Newton's polynomial.
  • Concerns are raised about the implications of publishing the formula, including fears of intellectual theft and questions about its originality.
  • Participants discuss the limitations of the formula, particularly regarding sequences defined by only a few terms, and the ambiguity in determining the next term without additional context.
  • One participant corrects their earlier formula expansion, indicating ongoing refinement of their claims.
  • Questions are posed about the nature of sequences and the intended next term, highlighting the complexities involved in defining sequences mathematically.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the proposed formula or its applicability to all sequences. Multiple competing views remain regarding the nature of sequences and the potential for a universal formula.

Contextual Notes

Limitations include the dependence on the number of terms provided in a sequence and the ambiguity in defining the next term based on limited information. The discussion reflects a range of interpretations and assumptions about sequences and their properties.

  • #31


If you carefully read the original post, you would see that there is no intrinsic flaw in his claim. He is not trying to create a nonsense formula for the nth term in a sequence given a list of the first k terms.

Instead he has a finite sequence u_1, ..., u_n. Given those numbers, he finds an explicit polynomial P(n) such that P(i)=u_i. There is no infinite sequence and it has nothing to do with pattern recognition. It is simply a polynomial interpolation and that is all he is claiming.

If the original post were not clear (which is understandable since the OP is not trained in maths), surely some of the follow up posts would have cleared this up??
 
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  • #32


Vargo said:
finite sequence
No such thing exists.
 
  • #33


johnqwertyful said:
No such thing exists.

While you're formally correct, it's a bit hairsplitting don't you think? We all know what he meant with it and many people do use this term.
 
  • #34


micromass said:
While you're formally correct, it's a bit hairsplitting don't you think? We all know what he meant with it and many people do use this term.

Not at all. I think it's important to use correct terminology; the way things are formally defined in a formal setting.
 
  • #35


johnqwertyful said:
Not at all. I think it's important to use correct terminology; the way things are formally defined in a formal setting.

It is formal. Many good math books use the terminology "finite sequence". For example: "Mathematical analysis" by Apostol, Definition 2.12.
 
  • #36


Or "Real Analysis" by Royden, page 9.

I'm sure I can find many others.
 
  • #38


i don't think so. i have one in mind whose nth term is ?
 
  • #39


Vargo said:
If you carefully read the original post, you would see that there is no intrinsic flaw in his claim. He is not trying to create a nonsense formula for the nth term in a sequence given a list of the first k terms.

Instead he has a finite sequence u_1, ..., u_n. Given those numbers, he finds an explicit polynomial P(n) such that P(i)=u_i. There is no infinite sequence and it has nothing to do with pattern recognition. It is simply a polynomial interpolation and that is all he is claiming.
The formula in the first post is not the one OP had in mind - that was posted later. That didn't help matters.
If the original post were not clear (which is understandable since the OP is not trained in maths), surely some of the follow up posts would have cleared this up??
Yet so many people are responding as if the claim is that the formula is predicting the next term in the intended sequence. Like:
mathwonk said:
i don't think so. i have one in mind whose nth term is ?
... I don't think OP is actually claiming to be able to predict the next term in mathwonk's sequence ... just to be able to come up with "a" next term given what has gone before and the assumptions (reasoning?) built-in to the formula. The formula would be predictive for a particular kind of sequence.

The wording in post #1 is unclear on this point, and muddied, rather than clarified, by crossed wires in later responses, but I think we are at the point where OP can be encouraged to confirm or deny the interpretation unambiguously.

I submit: further discussion is pointless until that happens.
 
  • #40


Simon Bridge said:
The wording in post #1 is unclear on this point, and muddied, rather than clarified, by crossed wires in later responses, but I think we are at the point where OP can be encouraged to confirm or deny the interpretation unambiguously.

I submit: further discussion is pointless until that happens.

He already did so, see the start of post #17. This thread is done, and the original poster said so himself in post #20.
 
  • #41


From post #20:
I didn't understand the formulas on the Wikipedia page that you posted here, but now I do. I see that this discussion could have ended a lot sooner.
... not specific.
The confusion still continued though: witness the remaining posts.

I agree the thread is finished with no further input from OP.
 
  • #42


Locked.
 

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