TITLE: nth term of sequences
QUESTION [8 upvotes]: 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?
Thanks!

REPLY [9 votes]: Your $27$ should be $24$. Now it's easy to see the pattern!
Edited to add:
To see why, take (for example) $a_n=n^4$. The first difference is  
$a_{n+1}-a_n = (n+1)^4 - n^4 = 4n^3 +$ lower powers of $n$  
So the fourth difference is four times the third difference of $n^3$ plus the third difference of lower powers of $n$. Lower powers have zero third difference, so this is $4 \times 6 = 24$.