With only two points and no further details, this sequence could take a variety of forms. Mark already showed you how easily you can calculate the arithmetic sequence. I noticed, as I glanced at your values, that it forms a very tidy geometric sequence as well. So, just in case that is what your teacher wanted, I decided to chime in with this option. That way, Mark and I have you covered.
The geometric sequence covers situations where, instead of adding the same thing every time (or subtracting, but we think of that as adding a negative), you instead multiply by the same factor. That creates this formula:
An = A0 * rn-1
In your case, the initial value, A0, wasn't provided, but we can find it without too many extra steps. First plug in what you do know:
A5= A0 * r5-1 A2 = A0 * r2--1
768 = A0 * r4 12 = A0 * r1
There are a couple of perfectly legal approaches you can take next, but I find this one easier for most students to follow. We simply divide one equation into the other:
768 = A0 * r4
/ 12 /A0 * r1
64 = r³
The A0 cancelled out. Now, we take the cube root, and get:
4 = r
We can plug this into one of the original equations to get A0
12 = A0 * 4¹
12 = 4 A0
/4 /4
3 = A0
The geometric sequence formula would become:
An = 3 * (4)n-1
Now you have the two most common sequence types covered in your class. One of them will be the answer your teacher sought.
