Any sequence of numbers Ak = f(k), where f(k) is a polynomial of degree n, will have its nth differences a constant. This is the method of “finite differences”; in fact, taking the differences of a sequence of numbers is roughly analogous to differentiation, and taking partial sums is analogous to integration.
It’s an interesting fact about the way mathematics has historically developed that the analogous statement about polynomials viewed as functions of real numbers seems much more obvious to most people that have some mathematical training.
Any sequence of numbers Ak = f(k), where f(k) is a polynomial of degree n, will have its nth differences a constant. This is the method of “finite differences”; in fact, taking the differences of a sequence of numbers is roughly analogous to differentiation, and taking partial sums is analogous to integration.
It’s an interesting fact about the way mathematics has historically developed that the analogous statement about polynomials viewed as functions of real numbers seems much more obvious to most people that have some mathematical training.