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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.

enter image description here

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?

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  • $\begingroup$ Yes you need to evaluate the changing area of a step $n$ to the step $n+1$. Start by calculating it for $n=1,2\dots$ $\endgroup$ – Atmos May 15 '18 at 19:07
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    $\begingroup$ Is it not the area of the paper itself? $\endgroup$ – Math Lover May 15 '18 at 19:10
  • $\begingroup$ At the beginning we have the root which has area equal to 1/3 of the sheet. Then the two branches have are equal to (1/3)/3 = 1/9 of the sheet. Is this correct so far? @Atmos $\endgroup$ – Mary Star May 15 '18 at 19:10
  • $\begingroup$ Is this maybe something like $\sum_{i=0}^n\frac{1}{3^i}$ ? @Atmos $\endgroup$ – Mary Star May 15 '18 at 19:13
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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

$$ S_0+\frac{2}{3}S_0+\left(\frac{2}{3}\right)^2S_0 +\cdots = S_0\sum_{k=0}^{\infty} \left(\frac{2}{3}\right)^k = 3S_0 $$

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    $\begingroup$ At the step $n$ from a sheet with area $S_n$ we generate an additional area $\frac{2}{3}S_n$ $\endgroup$ – Cesareo May 15 '18 at 20:00
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    $\begingroup$ From a sheet with area $S_n$ we generate two additional sheets with $\frac{1}{3}S_n$ area each. Each leaf generates two other leaves. $\endgroup$ – Cesareo May 15 '18 at 20:15
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    $\begingroup$ Yes. It is the DIN A4 area. $\endgroup$ – Cesareo May 15 '18 at 20:44
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    $\begingroup$ The total sum sum is $S_0+\frac{2}{3}S_0+\left(\frac{2}{3}\right)^2S_0 +\cdots $ $\endgroup$ – Cesareo May 15 '18 at 20:58
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    $\begingroup$ @MaryStar I took a sheet of old school graph paper and literally graphed the first three cases and then looked each time and the total area. I suggest you do the same. It will become incredibly clear! $\endgroup$ – imranfat May 16 '18 at 15:51

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