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It is impossible.
Given any set of n numbers, it is easy to find infinitely many polynomials of order n that can be used as a rule for those numbers. In addition, there are non-polynomial solutions. Short of reading the mind of the person who posed the question, there is no way of determining which of the infinitely many solutions is the "correct" one. The best that you can do is Occam's test: go for the simplest solution.
For example, what is the rule for the sequence 10, 20, 30, 40, 50?
Is it t(n) = 10*n?
Why not t(n) = (n^5 - 15*n^4 + 85*n^3 - 225*n^2 + 334*n - 120)/6 ?
For n = 1 , 2, 3, 4, 5 they give the same result: the sequence for which you are seeking a rule. Mathematically, both answers are equally valid.
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What is the rule for this pattern 2 6 8 14?
Without further terms in the sequence, it is impossible to determine what the rule in the sequence is.
What is the rule in generating a sequence?
Anything you like. You specify whatever rule you like and the resulting set of numbers is the sequence based on that rule.
What make a number sequence?
A number sequence is an ordered set of numbers. There can be a rule such that the next number in the sequence can be determined by the values of some or all of the preceding terms in the sequence. However, the sequence for a random walk illustrates that such a rule is not necessary to define a sequence.
What is the rule for this sequence 12631.5?
A single number, such as 12631.5, does not make a sequence.
How do you find the rule of a sequence of numbers?
Since a given sequence of numbers can be designed to follow any rule, you have to use a system of trial and error to see if you can discover the rule. Sometimes the rule is obvious, sometimes it is extremely complicated. Try to invent a rule which would produce the sequence that you observe.
