A DIN A4 sheet is divided into thirds. A rectangle is the root of the tree, the other two rectangles are each divided into thirds again. Two rectangles form the branches - one to the left, one to the right - the others are again divided into thirds and so on.
I want to calculate the area of that tree.
Do we have to write a formula using series or a recursive formula? But how exactly? Could you give me a hint?
Each sheet gives origin to two other sheets according to
$$ S_n \to \frac{2}{3}S_n $$
so the total area is given by
$$ S_0+\frac{2}{3}S_0+\left(\frac{2}{3}\right)^2S_0 +\cdots = S_0\sum_{k=0}^{\infty} \left(\frac{2}{3}\right)^k = 3S_0 $$