## Modeling Sequences with Polynomials

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

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.