What is the result of sum $\sum\limits_{i=0}^n 2^i$ 
Possible Duplicate:
the sum of powers of $2$ between $2^0$ and $2^n$ 

What is the result of 
$$2^0 + 2^1 + 2^2 + \cdots + 2^{n-1} + 2^n\ ?$$
Is there a formula on this? and how to prove the formula?
(It is actually to compute the time complexity of a Fibonacci recursive method.)

Let $S = 2^0 + 2^1 + 2^2 + \cdots + 2^{n}$.
Then $2S = 2^1 + 2^2 + 2^3 + \cdots + 2^{n} + 2^{n+1}$.
Then
$$\begin{align*}
S = 2S - S &=     & & 2^1 &+& 2^2 & + & 2^3 & + & 2^4 &+&\cdots &+& 2^{n} &+& 2^{n+1}\\
&& -2^0 -& 2^1 & - & 2^2 & - & 2^3 & - & 2^4 & - & \cdots & - & 2^n
\end{align*}$$
How much is that?
