TITLE: Interpreting images representing geometric series
QUESTION [5 upvotes]: I understand the formula for infinite geometric series as 
$$S = \frac{a_{1}}{1-r}$$ if $0<r<1$
However I'm having trouble applying it to these images 
It seems to me that in the first image, the first square represents 1/4 of the entire square
For the second and third images, the respective rectangle and triangle make up 1/2 of the entire square. 
Not sure what to do with this. Does it mean that for the first image for example, the image is $$\sum_{n=0}^\infty \left(\frac{1}{4}\right)^n$$

REPLY [2 votes]: You can solve these without geometric series. 
Say the proportion of coverage of the first image is $s$. Split the first image up into four squares. The topleft and bottomright are not covered at all. The bottomleft square is fully covered. And the topright square? Well it's covered in exactly the same way as $s$ itself. So we find:
\[s = \frac{1}{4}(0 + s + 1 + 0)\]
\[4s = 1 + s\]
\[s = \frac{1}{3}\]
Similarly for the other images we find equations $s = \frac{1}{4}(1 + s + 1 + 0)$ and $s = \frac{1}{4}(\frac12 + s + 1 + \frac12)$.