How to prove that $2^0 + 2^1 + 2^2 + 2^3 + \cdots + 2^n = 2^{n+1} - 1$ When I'm reading Computer Systems: A Programmer's Perspective, I met the sum of binary numbers and failed to prove it:
$$ 2^0 + 2^1 + 2^2 + 2^3 + \cdots + 2^n = 2^{n+1} - 1 $$
This might be preliminary knowledge, I'm not good at mathematics, any body could give me a hint?

Notice $2^0=2^1-1$.
Then $2^0+2^1=2(2^1)-1=2^2-1$
Then $2^0+2^1+2^2=2(2^2)-1=2^3-1$.
We have $2^0+2^1+2^2...2^{n-1}=2(2^{n-1})-1=2^n-1$.
And to show the this holds for the $n+1$ case, notice that $(2^n-1)+2^n=2(2^n)-1=2^{n+1}-1$.
