I found these on the Art of Problem Solving website here.
I just don't see how the first image would give you $\frac{1}{3}+\frac{1}{3^2}+...$ and I'm unable to see how the second one is divided into fourths. Normally for these problems, I can look at the biggest picture and see how many squares are a certain color and notice the pattern as you increase the number of squares but I just can't see the pattern. Can someone please explain?
Assume that the biggest square is a unit square. The first $3\times 1$ rectangle has an area $\frac{1}{3}$. The small red square in the first row and the second column has an area $\frac{1}{3^{2}}$. The second smaller $3\times 1$ rectangle has an area $\frac{1}{3^{3}}$, and so on. So the total area of the red part is $$ \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \cdots $$ and the same for the green part. Now, the sum of red and green parts is 1, which is the area of the biggest unit square.