solving a problem with induction The original question is the following


  prove that $2·\sum_{i=0}^{n-1} 3^{i} = 3^n-1$ for all n $\geq$ 1


I know that I have to prove by induction and have successfully done the base case, my IH is the following: 

Assume $2·\sum_{i=0}^{n-1} 3^{i} = 3^n-1$ holds for all n $\geq$ 1 to prove $2·\sum_{i=0}^{n} 3^{i} = 3^{n+1}-1$

then I did this:
$2·\sum_{i=0}^{n} 3^{i} = 2·\sum_{i=0}^{n-1} 3^{i} +3^n$ 

then by my IH: $2·\sum_{i=0}^{n-1} 3^{i} +3^n = 3^n-1+3^n $

after this I struggle to continue, I know I have to find a way to rewrite it to $3^{n+1}-1$ and I know this is equal to $3·3^n-1$ but I don't see how to get from the one to the other. Anyone who can help?  

edit: I mistook a 2 for a 3, can anyone help now?
$2·\sum_{i=0}^{n} 3^{i} = 3^n-1 +2\cdot{3^n}=3^{n+1}-1$.