I'm a Phd student who teaches part time at a high school and I noticed something when teaching sequences today. I asked my students to find the nth term (the general term) for some sequences. They observed:
If $a_n=n$ then the first differences will be $1$.
If $a_n=n^2$ then the second differences will be $2$.
If $a_n=n^3$ then the third differences will be $6$.
If $a_n=n^4$ then the fourth differences will be $27$.
So I can now construct a sequence: $1,2,6,27,120,720,...$
What are these numbers? Can I find $a_n$? How?