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

$$2^{10} + 2^{9} + 2^{8} +2^7+2^6+2^5+2^4+ 2^{3} + 2^2 + 2^1 = 2^{11} - 2$$
Add $(2-2)$ to the left-hand side, obtaining:
$$\begin{align}
2^{10} + 2^{9} + 2^{8} + 2^7+2^6+2^5+2^4 + 2^{3} + 2^2 + 2^1  \color{green}{+ 2  - 2} & = 2^{11} -2 \\
2^{10} + 2^{9} + 2^{8} + 2^7+2^6+2^5+2^4 + 2^{3} + 2^2 + \color{maroon}{2^1 + 2^1}  - 2  & = 2^{11} -2 \end{align}$$
Since $2^1 + 2^1 = 2\cdot(2^1) = 2^2$, the left side simplifies to:
$$2^{10} + 2^{9} + 2^{8} +2^7+2^6+2^5+2^4+ 2^{3} + \color{maroon}{2^2 + 2^2} - 2 = 2^{11} - 2$$
Since $2^2 + 2^2 = 2\cdot(2^2) = 2^3$, the left side simplifies to:
$$2^{10} + 2^{9} + 2^{8} + 2^7+2^6+2^5 + 2^{4} + \color{maroon}{2^3 + 2^3} - 2 = 2^{11} - 2$$
Repeat this simplification several more times:
$$2^{10} + 2^{9} + 2^{8} + 2^7+2^6+2^5  + \color{maroon}{2^4 + 2^4} - 2 = 2^{11} - 2 \\
\\ \vdots \\\color{maroon}{2^{10} + 2^{10}} - 2 = 2^{11} - 2\\
\color{maroon}{2^{11}} - 2= 2^{11} - 2$$
