# Giving recursive definition

I need to give the recursive function of $3n^2$. I'm pretty sure the base case needs to be $3 \cdot 0^2 = 0$, but I don't know where to go from there.

Any help is appreciated.

If $a_n=3n^2$, then $$a_n-a_{n-1}=3n^2-3(n-1)^2=6n-3$$ Therefore, $$a_n=6n-3+a_{n-1}$$ $$a_0=3\cdot 0^2=0$$ is the recursive relation.

• Is this the same as a recursive definition? My text book talks about first creating a base case and then a second part that eventually will use the base case. Maybe that doesn't apply to this problem though.
– Josh
Dec 3, 2015 at 1:26
• @Josh Yes you need a base case, since otherwise the recursive relation will keep going down to $-\infty$. You correctly stated in your question that $a_0=0$. Dec 3, 2015 at 1:27
• Ah okay, sorry I'm a little confused here. So the entire definition would be my base case and then your relation?
– Josh
Dec 3, 2015 at 1:28
• @Josh Indeed, both parts are necessary for a full recursive formula. I updated the answer to clarify this. Dec 3, 2015 at 1:29
• Great, I understand now, thanks.
– Josh
Dec 3, 2015 at 1:30