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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?

user130306
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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.

Seewoo Lee
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  • so we go from 3 squares (in the 3x1) to 1 square to 1/3rd of 1 square, etc? i understand the whole thing is the unit square and then you take one third of it but then do you take only 1/9th of it and then you take 1/3*(1/9)? – user130306 May 12 '19 at 04:25
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    @user130306 So it is like a sum of rectangle + square + rectangle + square + ..., which is given as the above sum. – Seewoo Lee May 12 '19 at 04:35
  • ohh thank you, that makes sense. – user130306 May 12 '19 at 06:23