Why is $2^b-1=2^{b-1}+2^{b-2}+...+1$? Can I get an intuitive explanation why the formula in the title holds?
I know that it works but I am not sure why
$2^b-1=2^{b-1}+2^{b-2}+...+1$

You can visualize it by looking this identity as identity of polynomials: 
$$
(1-x)(1+x)=1-x^2
$$
$$
(1-x)(1+x+x^2)=1-x^3
$$
and in general 
$$
(1-x)(1+x+x^2+x^3+\ldots+x^{n-1})=1-x^n
$$ Now on substituting $x=2$, we get the desired result. Hope it helps.
