I’m teaching an after school workshop for a few 7th graders. I was having them try to predict the next item in a complicated sequence. After some failed attempts, one of the kids started analyzing the differences between the numbers, then the differences between those differences, and used that to predict the next few items on the list. (This worked because the list was generated by a quadratic function but they don’t know this part; they can predict the next list items but cannot write a formula yet).

I’m trying to assess the students and am not sure whether they just discovered that technique or had learned it in school already. In what grade are sequences and common differences usually covered? I’m talking about in the US, specifically NYC.

I’m also interested in pushing the idea further with them, but ideally in a different direction form what will be done in school. How often does the following concept get covered in MS-HS, and when?

Given a sequence $S=(s_1,s_2,s_3,dots)$,

its sequence of common differences is

$D(S)=(s_2-s_1,s_3-s_2,s_4-s_3,dots).$

Now take a polynomial

$$f(n)=c_kn^k+c_{k-1}n^{k-1}+dots+c_0,$$

and form the sequence

$F=big(f(0), f(1), f(2), dotsbig)$.

Then $D^k(F)$

is a constant string of the number $c_kk!$,

and using this one can determine the polynomial completely.