So the questions is asking me to come up with a formula for the following sequence.
8, 16, 32, 64, …
The answer they give is $A_n = 2^{n+2}$. When I looked at it, I saw $A_n = A_{n-1} \cdot 2$.
Is there something that makes my answer wrong? Is there a specific formula I should see when dealing with geometric sequences?
On
Basically, we have our series as $G = 8,16,32,\cdots $. We have the first term as $8$ and the common ratio $r=2$ We now write the general term of this sequence $G $ as like what you have said with some manipulation is: $$G_n =8\times 2^{n-1}=2^3\times 2^{n-1}= 2^{n+2} ...(1)$$. We also get $$G_{n+1} = 2^{n+3}...(2) $$. Thus, we may express the $n+1$th term of $G $ in terms of the $n $th term as $$G_{n+1} =2G_{n}...(3) $$ But to the best of my knowledge the general term of a G.P is expressed as $(1) $ even though $(3) $ can be brought from $(1) $ as dxiv says in his comment by telescoping. The two forms are thus very much identical but just expressed in different ways.
Your formula is correct, but when only giving the recursion formula you need to define a starting point $A_0$. $A_0=4$ in your example, since the sequence $6, 12, 24, ...$ also is of the form $A_n=2\cdot A_{n-1}$, but with a different $A_0=3$.
Then you can also express your $A_0$ as $2^a$, in your example $a=2$ since $2^2=4=A_0$ giving the explicit formula.