## Modeling Sequences with Polynomials

November 11, 2010 at 20:10 (math.RA, mathematica)

Suppose I asked you to find the next term in the sequence

2, 4, 6, 8, 10, …

Of course the expected answer is 12. But then I could tell you that these terms correspond with the formula

$f(n) = -2 + \frac{197}{30}n - \frac{15}{4}n^2 + \frac{17}{12}n^3 - \frac{1}{4}n^4 + \frac{1}{60}n^5$

so the actual sequence has

2, 4, 6, 8, 10, 14, 26, 58, 130, 272, …

This illustrates a common misconception that there is a unique solution to these sorts of problems. In fact, starting with the first five evens, we can pick any real number as the sixth term and find a polynomial of degree at most 5 to model it using some basic linear algebra.