mystery function

Question

How does one find an equation for data given a set of ordered pairs (the function is not linear)?

My series begins: (1,1), (2,5), (3,14), (4, 30), (5, 55), (6,91).

I am curious about the answer but even more, I'm curious about the process of finding the answer.

Answer 1

The answer is f(n) = $\frac{(2n^3+3n^2+n)}{6}$ = (1/6)n(2n +1 )(n + 1).

How to find it?

Make a difference table.

You should immediately recognize the first differences as the sequence $(n+1{)}^2$ for $n\ge 1$ . This should tell you that the original sequence is a cubic. The only way I know to find the original sequence involves knowledge of the calculus of finite differences, which I cannot explain here. Perhaps your instructor will show you a different way.