Can you tell me how to find logically a formula if series of numbers is given. for example $$2,4,8,16$$ the formula for this series is $2^n$ where $n$ is the $n$-th position. This is an easiest series but lets say a complex series has been given (eg:$0, 12, 10, 0, –12, –20,$) how to find its formula? Advance thanks for your help.
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One useful resource is checking the On-Line Encyclopedia of Integer Sequences (OEIS). For many sequences of general interest, you will find details there, often including one or more formulas where such a formula is known. In your second example, there is no match, but instead a suggestion that this might be
$$x^3 − 13 x^2 + 44 x − 32$$
If you are interested in finding a polynomial to describe a given sequence of numbers, check out polynomial interpolation.
If not all formulas are polynomials, then you'll have to provide more information about the possible forms of the formulas you're interested in, or about some other definition of the sequence, perhaps using some recursive equation.