Say you have two sequences which are given by a polynomial of some degree. Both of these sequences share a term. Is there a way to find the missing term, and what is the minimum number of values (or information in general, such as degree of the polynomial) for each sequence for a unique solution to be found?
To make this a little bit clearer, I have an example of sequences which I know share a term but I do not know what that term is. The sequences are:
- 0, 6, 30, __
- 0, 12, 40, __
I also know that both of these sequences are increasing and that they may be third degree polynomials (they definitely are the same degree).
From the information given by these two sequences, is it possible to find out what the forth term (which is common to both of them) is, and how do you go about solving such a problem?