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

1. Feb 20, 2013

### karpmage

Hi there,

I recently discovered a formula for the nth term that works for any finite sequence of numbers. I was just wondering whether a formula has already been discovered, and if not, how and where I should publish it.

To give you an example of what i mean:

one formula for the nth term of any sequence with two numbers is:
(Un)=(u1)+((u2)-(u1))(n-1)

I've tried googling a formula but have come up with nothing. I asked my maths teacher and he didn't know either. He recommended that I consider publishing it.

(I'm worried that if I post the formula on this website, someone might steal it and I won't get credit for its discovery (if it is in fact my discovery). I'd also like your opinion on this.)

Thanks.

2. Feb 20, 2013

### jbunniii

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

This isn't a formula for an arbitrary sequence, only an arithmetic progression, i.e. one which adds a fixed number at each step, such as 3, 8, 13, 18, 23, 28, ... Of course it's well known, almost trivial. Most high school algebra books mention it at some point.

There can't be any formula for predicting an element of a general sequence in terms of the other elements. Proof: I'll tell you the other elements, you tell me what number your formula predicts for the remaining one, and I'll spoil your day by picking a different number.

3. Feb 20, 2013

### karpmage

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

I used this formula as an example as I knew it was very well known. You're right when you say that you can pick a different number. If you're given any sequence of numbers and asked to find the next number (and the sequence isn't defined as arithmetic, geometric, etc.), you can give any number and you will still be correct, provided that you have a formula to back it up.

4. Feb 20, 2013

### jbunniii

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

My point is that your formula can't work for all possible sequences. So what class of sequences does it work for?

5. Feb 21, 2013

### pwsnafu

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

Sounds like polynomial interpolation.

You can always find such a formula: see Newton polynomial.

6. Feb 21, 2013

### CompuChip

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

Or I can write down a formula stating that the elements of the sequence are equal to the N numbers that I gave you for n <= N and to some arbitrary other sequence not matching your prediction for n > N.

Anyhow, I think we are now misinterpreting the question. For example, I can give you {1, 1} as the first two elements. How will you know if I'm talking about {1, 1, 1, 1, 1}, {1, 1, 2, 3, 5} or {1, 1, 2, 2, 3}?

7. Feb 21, 2013

### karpmage

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

From what I can see, there's no real point publishing what I've found. So I'm going to post my formula here.

The formula takes the form of the binomial expansion, where:

un= (u-1)0/0! + (u-1)1*(n-1)/1! + (u-1)2*(n-1)(n-2)/2! + (u-1)3*(n-1)(n-2)(n-3)/3! + ...

you then need to expand the sequence and replace un with un+1 (for example u3 gets replaced with u4. Only "un" gets replaced by un)

This will give you:

un= u1 + (u2-u1)(n-1)+(u3-2u2+u1)(n-1)(n-2)/2 + (u4-3u3+3u2-u1)(n-1)(n-2)(n-3)/6 + ...

Which works for any given sequence of numbers.

I'm not sure whether this is the same as what you guys have posted.

Last edited: Feb 21, 2013
8. Feb 21, 2013

### micromass

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

Sorry man, you were beaten by the 6 year old Gauss: http://www.education2000.com/demo/demo/botchtml/arithser.htm [Broken]

Last edited by a moderator: May 6, 2017
9. Feb 21, 2013

### karpmage

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

the issue isn't with the formula, it's with the number of numbers in the sequence. As I said, this gives a formula for the nth term for any sequence where only the first two terms are given. Of course, a sequence with only two terms could be arithmetic, or geometric, or something else altogether. With the information given, it's impossible to determine the next term, but a formula for the nth term can be found.

10. Feb 21, 2013

### karpmage

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

it doesn't have to be an arithmetic series. The formula that I displayed in my first post isn't the one that I'm talking about. The formula that you've shown isn't a formula for the nth term, at least not that I can see.

Last edited by a moderator: May 6, 2017
11. Feb 21, 2013

### johnqwertyful

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

Sequences, by definition, are infinite.

Also "impossible to determine the next term" but somehow you can find the nth term? What about n=3?

12. Feb 21, 2013

### karpmage

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

Sorry, my formula was wrong because I expanded (u-1)3 wrong. It's corrected now.

13. Feb 21, 2013

### karpmage

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

That's more of a problem with sequences in general. You can say that about any sequence. For example, how would know, with absolute certainty, if {1,2,4,8} leads to {1,2,4,8,16,32} from the formula 2n or {1,2,4,8,15,26} as my formula states (by stopping to input values at a point such that it is a cubic. In actual fact, you could say that it is a quintic and put whatever values you want in.)

14. Feb 21, 2013

### pwsnafu

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

This is what I think you are claiming:

We have some finite list of numbers, for example {1,3,5,7,11,13}. We substitutes these numbers into the expression I quoted, u1 = 1, u2 = 3, etc. Then the right hand side is some expression in the variable n. If we set n = 3, for example, the right hand side evaluates to 5.

Is this correct? Are you claiming anything more? Why did your teacher suggest you should publish?

Last edited: Feb 21, 2013
15. Feb 21, 2013

### karpmage

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

Let me rephrase, it's impossible to determine what is the intended next term, or intended formula for the nth term (except by luck) because there is an infinite number of terms that could take the place of n=3. Essentially, there is no need to do any calculations when determining the next term, since any answer is correct. This kind of makes questions like "Ex [2] The next term of 5, 11, 17, 23,... is ________." (http://www.math-magic.com/sequences/next_term.htm) pointless, unless it is explicitly stated to be an arithmetic or geometric, etc. sequence. One example where you could put in any answer is "The next term of 2, 5, 14, 41,... is _______." Does this mean that examiners can't mark these questions incorrect? I'd like to know.

I also said that you can find "A" formula, rather "the intended" formula

Last edited: Feb 21, 2013
16. Feb 21, 2013

### pwsnafu

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

In mathematics? There is no wrong answer.

In an IQ test? It's to test whether you can identify patterns so there is a right answer.

17. Feb 21, 2013

### karpmage

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

This is all I'm claiming. When he said that he suggested that I publish, I think he was talking generally, as I spend alot of time thinking about stuff like this. He didn't look into too much detail when I showed him either. He didn't actually check up on whether it was real or not either, he just didn't know off of the top of his head, so I probably shouldn't have put that in. Is this basically the same thing as polynomial interpolation/linear interpolation/newton polynomial? Did they have a general formula?

18. Feb 21, 2013

### karpmage

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

These type of questions came up in my igcse exam. They were phrased similarly to the ones I posted. I remember one of them gave a sequence of 5 or 6 numbers, and then asked for the next term and a formula for the nth term. It was something like 3*((2^n)-1). However my sequence would also have given the correct number. As for the ones that didn't ask for the next number, would I have been incorrect if I didn't give the intended answer, but explained why it still works?

19. Feb 21, 2013

### karpmage

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

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

I just found this. The segment on Newton Series is pretty much exactly the formula that I put down, just phrased a little differently. I'm going to have to concede that Newton beat me to it.

20. Feb 21, 2013

### karpmage

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

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.

21. Feb 21, 2013

### micromass

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

Let's say you are given 2,3,5,7,11,13,... (= the primes) How would you determine the next number using your method? If you can do this, then you have discovered a formula for the sequence of prime numbers.

22. Feb 21, 2013

### Simon Bridge

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

That is the claim under discussion.
When considering the replies, you should realise that this is a specific claim in mathematics ... a mathematician will read this to mean that the discovered formula can predict the next number in the series.
This comes up a lot - one of the advantages of making a discovery public is that lots of people witness you as the originator of the discovery. In the event of a dispute, you'll have a record of exactly when and how you made it first. If you keep it secret, then someone else may beat you to the punch, and then you have no proof you got there first.

You certainly should not tell people you've made an important discovery until you are prepared to tell them what it is.
Which is why your formula does not satisfy the claim you made in post #1 (quoted above). The general nature of sequences defeats you. It can, however, work for a particular class of sequences. That is, if it is supposed to be predictive... but I suspect the following quote shows that you don't mean to predict the actual next number ... i.e. if I fed you the sequence 3141 - the formula would give you "a" next number consistent with these, but probably won't continue 5926 (the sequence of digits in pi).
Depends on what is being examined.

Exams are to be taken in the context of the material that is being examined and not just the literal content of the exam paper. The student is expected to make a judgement about what is intended - that is part of the test.

I suspect you mean that you have found a systematic way to generate a number that is consistent with the sequence of numbers that have come before. i.e. out of the infinite possible continuations of a sequence of numbers, this formula is a way of picking one.

That is not the same as a formula for finding the next number in a sequence ... at least not the way mathematicians understand those words. To illustrate this - consider the sequence (suggested by micromass) of primes.

The formula would generate "a" number for the next in the sequence, consistent with the previous ones (and the assumptions that went into the formula), but it probably won't be a prime number. The additional information that it is has to be a prime number makes all the difference.

Back to the exam situation - it is the context of the exam question that restricts the kinds of answers that will be considered correct.

eg. If it is a pattern-recognition test in math, then 31415 may be recognized as the sequence of digits in pi ... so the next three numbers would be 926, but a puzzle from MENSA would suggest that we should be suspicious of such an easy answer and look further, like the sequence of chimes on a clock that chimes once each half-hour... so the continuation would be 161... thogh that would be a fairer puzzle if an earlier sequence were chosen like 19110111112111213 ...

This help?

23. Feb 21, 2013

### micromass

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

24. Feb 21, 2013

### Simon Bridge

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

Aw shucks, twern't nothin.
I will agree - provisionally. However: pattern recognition is often given in many countries as part of a math course (pre-secondary usually).

The course, so early on, usually has not covered many possible series so the test is just seeing if students can recognize them. When they are encountered, especially later, there is always some metadata to be taken into account. Otherwise it's meaningless.

After a while, students learn to state their assumptions explicitly - and it's a harder lesson that that sounds. We all make many assumptions without realizing them. I suspect OP needs to start doing that.

These exercises can be useful as a precursor to mathematical modelling as in fitting curves to data or projecting a trend from previous data. Empiricism yes? The impossibility of finding the next in sequence from the previous ones by themselves is pretty much a form of the philosophical "problem of induction" right? So we make a bunch of assumptions and hold the conclusions provisional.

I'm interested in how OP understands all this ...

25. Feb 21, 2013

### micromass

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

Of course pattern recognition is an extremely important skill in mathematics. If you're not good in pattern recognition, then you won't be a very good mathematician, or scientist.

My statement only covered IQ tests and pattern recognition tests. They are not math in the sense that the answer isn't well-defined. There is no real "correct" answers. Or rather: there is no definition for what the correct answer should be. Or: there is no algorithm for checking whether your answer is the right one.

It is of course entirely possible to make pattern recognition tests which are mathematical (that is: for which there is a good answer). In fact, most of the exercises on mathematical induction tend to rely on pattern recognition. Of course, the crucial part is that we can also actually check our answer. If we can do that, then it's math.