Why is it that $2^{10} + 2^{9} + 2^{8} + \cdots + 2^{3} + 2^2 + 2^1 = 2^{11} - 2$? Why is it that $$2^{10} + 2^{9} + 2^{8} + \cdots + 2^{3} + 2^2 + 2^1 = 2^{11} - 2?$$

Let $S = 1+2^1+2^2+...+2^{10}$
Multiply by 2
$2S = 2^1+2^2+2^3+...+2^{11}$
Substract the former from the latter:
$S = 2^{11}-1$
$1+2^1+2^2+...+2^{10} = 2^{11}-1$
