What should I read up on for techniques on generalizing a sequence to find the nth term?
An extremely simple example:
1, 3, 7, 15...
A formula could be (2^n) - 1
Thanks in advance!
PS. Sorry if this question is too vague!
Printable View
What should I read up on for techniques on generalizing a sequence to find the nth term?
An extremely simple example:
1, 3, 7, 15...
A formula could be (2^n) - 1
Thanks in advance!
PS. Sorry if this question is too vague!
This is a common question, apparently designed to make the poser of the question appear smart in proportion to the subtlety of the purported relationship among the terms. In fact it demonstrates a fundamental lack of understanding of the concept of a function on the part of he who poses the question..Quote:
Originally Posted by m84uily
A sequence is nothing more and nothing less than a function, the domain of which is the positive integers.
Thus
1,3,5, 7, 9 , .... (odd numbers)
1,3,5, 7, 13, ... (prime numbers)
1,3,5,7, brown cow, yellow house, etc. (random junk)
are all sequences.
Knowing the first several terms of a sequence tells you nothing about the next term or any subsequent term
in case you dont know what the generalization of the nth term for this sequence is, its:Quote:
An extremely simple example:
1, 3, 7, 15...
A formula could be (2^n) - 1
NO!Quote:
Originally Posted by Heinsbergrelatz
That is A "generalization", not THE generalization. Knowing the firsy 4 terms in a sequence tells you nothing about the remaining terms.
Yes indeed, my bad ..Quote:
Originally Posted by DrRocket
I don't think I worded my question properly, sorry about that! I'll try again:Quote:
Originally Posted by DrRocket
What should I read up on for techniques to find order among a list of terms to attempt to predict future results?
Here's another example to better explain what I mean:
0,1,4,9,16
d1 = 1,3,5,7
d2 = 2,2,2
The second difference is constant and tells us that this could be modeled by a second degree polynomial.
an^2 + bn + c
n = 0
0a + 0b + c = 0
c = 0
n = 1
a + b = 1
a = 1 - b
n = 2
4a + 2b = 4
4(1-b) + 2b = 4
4 - 4b + 2b = 4
-2b = 0
b = 0
a = 1
n^2
Unless you're telling me that no such thing exists! In which case, sorry that somewhat went over my head.
Well first you are using the word "order" in way different what I (and I suspect DrRocket) would.Quote:
Originally Posted by m84uily
Just to reinforce what DrRocket said: Suppose we denote the set of all positive integers by . Then a sequence is the mapping where S can be any set whatever (even the Cartesian product of sets, but that's grown-up stuff).
So the image point is by convention denoted , this being the n-th element in some sequence
This works because the positive integers have a "natural" ordering which our function "imposes" on the codomain S according to some rule. The codomain S is not required to have have any sense of order of its own.
By this I mean that the mapping (function) that assigns the consecutive indices to each is a "rule" that says how this is to be done. But given only part of the sequence, you can only guess this rule; which is no more than to say, given knowledge of the past, you can only guess the future - a common experience
I would like to learn techniques to guess rules for a codomain.Quote:
Originally Posted by Guitarist
Is that an okay way to word what I'm after?
The basic point here is that there is no such thing.
The word you are looking for is "extrapolation" and it rarely works well, but if you make some assumptions, you can get a guess. For example, if you assume you're dealing with a polynomial, you can take the smallest-degree polynomial that fits the data and extrapolate from that. (See also the calculus of differences. The wiki page on that isn't very readable though.) If you assume it's some exponential function, you can take the quotient of each pair and then extrapolate from that.
Or you could always type your numbers in to the Online Encyclopedia of Integer Sequences (a catalog of interesting sequences). 1,3,7,15 returns 175 results BTW.
Okay, thanks a lot!Quote:
Originally Posted by MagiMaster
Quote:
Originally Posted by m84uily
Give me the first n points of a sequence. I will give you a polynomial of degree n-1 that yields those points.
I can give you infinitely many higher degree polynomials that also yield those points.