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

Your sum is as follow:
$2(2^9+2^8+\cdots +2+1)$
which can be proved simply is equal to $2 \frac{2^{10}-1}{2-1}$ as desired.
