Here is a difference table for a function $S(N)$. Your job is to fill in the table, and then write out the closed polynomial expression corresponding to $S(N)$. The third level difference row is assumed to be a constant of $1$.
N 0 1 2 3 4 5
S(N) 2 - - - - -
∆ 0 - - - - -
∆2 3 - - - - -
∆3 1 1 1 1 1 1
I solved the table to get the following:
N 0 1 2 3 4 5
S(N) 2 2 5 12 24 42
∆ 0 3 7 12 18 25
∆2 3 4 5 6 7 8
∆3 1 1 1 1 1 1
But I'm completely lost on how to write out the closed form polynomial. Can someone get me started?
