# Calculate the area of the tree

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?

• Yes you need to evaluate the changing area of a step $n$ to the step $n+1$. Start by calculating it for $n=1,2\dots$ – Atmos May 15 '18 at 19:07
• Is it not the area of the paper itself? – Math Lover May 15 '18 at 19:10
• 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 – Mary Star May 15 '18 at 19:10
• Is this maybe something like $\sum_{i=0}^n\frac{1}{3^i}$ ? @Atmos – Mary Star May 15 '18 at 19:13

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

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

• At the step $n$ from a sheet with area $S_n$ we generate an additional area $\frac{2}{3}S_n$ – Cesareo May 15 '18 at 20:00
• 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. – Cesareo May 15 '18 at 20:15
• Yes. It is the DIN A4 area. – Cesareo May 15 '18 at 20:44
• The total sum sum is $S_0+\frac{2}{3}S_0+\left(\frac{2}{3}\right)^2S_0 +\cdots$ – Cesareo May 15 '18 at 20:58
• @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! – imranfat May 16 '18 at 15:51