General Formulas for Sequences

[Pupil]

Suppose I give you a sequence of numbers such as 2, 4, 6, 8, 10... and ask you to find the next integer. You would probably tell me 12, because the sequence follows the rule 2n where n is the ordinal number. But if I told you the next number in the sequence is 42, your rule wouldn't work, and you'd have to find a new one. Suppose you find this said rule, and I tell you, "nope," because the next integer is 8379356 and the your new rule won't work for this new sequence 2, 4, 6, 8, 10, 42, 8379356... But when we think about the original sequence I gave (2, 4, 6, 8, 10...), you found 3 general rules that would work, not just 2n! So my question is then: for any finite list of numbers in a sequence, is there always one or more than one general rule that will work for the series?

 

[boboYO]

yeah, you can always make a polynomial that will work:

http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html
quence
it follows that there an infinite number of 'rules' you can make up for any finite sequence