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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!

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1,2,6,24,120,720... Just guessing --> Isn't that the sequence of $n!$ for n = 1,2,3,4.... –  Roupam Ghosh Mar 4 '11 at 12:35
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up vote 9 down vote accepted

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 $4 \times 6 = 24$.

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yep thanks! $a_n=n!$ idiot. :) –  Jack Mar 4 '11 at 12:36
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