Why does this simplifies like that? 
I don't understand how do we simplify this fraction ? How does 3^n becomes 3^(n+1) and how does 1-(1/3) becomes 2 ???
Is there a general rule for this ?

The following is an explicit path from your statement to it's simplification:
$$\frac{3^n}{ (1 - \frac{1}{3}) }
= \frac{3^n}{(\frac{3}{3} - \frac{1}{3})} 
= \frac{3^n}{(\frac{3-1}{3})}
=\frac{3^n}{(\frac{2}{3})}
= 3^n\cdot (\frac{2}{3})^{-1}
= 3^n\cdot\frac{3}{2} 
= \frac{3^n \cdot 3^1}{2} 
= \frac{3^{n+1}}{2} $$
or generally, you could try the following:
$$1\cdot\frac1{1-\frac1{x}} = \frac{x}{x}\cdot\frac1{1-\frac1{x}} = \frac{x}{x(1 -\frac{1}{x})} = \frac{x}{x -1}$$
It doesn't matter how you simplify as long as you do it right and understand how to get the answer.
