## Precalculus (6th Edition) Blitzer

$\frac{1}{3}$
The shaded areas form a geometric series: $(0.5a)^2, ((0.5)^2a)^2, ((0.5)^3a)^2 ,...$ where $a$ is the side length of the largest square. We, have $a_1=(0.5)^2a^2, r= (0.5)^2$ Thus, the sum of all the areas is $S=\frac{a_1}{1-r}=\frac{(0.5)^2a^2}{1-(0.5)^2}=\frac{1}{3}a^2$ and the ratio is $\frac{S}{a^2}=\frac{1}{3}$