College Algebra, Chapter 9, 9.3, Section 9.3, Problem 80

A circular disk of radius $R$ is cut out of paper, as shown in figure (a). Two disks of radius $\displaystyle \frac{1}{2} R$ are cut out of paper and placed on top of the first disk, as in figure (b), and then four disks of radius $\displaystyle \frac{1}{4} R$ are placed on these two disks, as in figure (c). Assuming that this process can be repeated indefinitely, find the total area of all the disks.

If the area of the disk in part (a) is $\pi R^2$ and the area of the disks in part (b) is $\displaystyle 2 \pi \left( \frac{R}{2} \right)^2$. Then their common ratio is $\displaystyle r = \left( \frac{2 \pi \left( \frac{R}{2} \right)^2 }{\pi R^2} \right) = \left( \frac{\displaystyle 2 \pi \left( \frac{R^2}{4} \right) }{\pi R^2} \right) = \frac{1}{2}$. Therefore, the total area
of the disks will be

$\displaystyle S = \frac{a}{1 - r} = \frac{\pi R^2}{\displaystyle 1 - \frac{1}{2}} = \frac{\pi R^2}{\displaystyle \frac{1}{2}} = 2 \pi R^2$