20. Pythagoras Tree

A Pythagoras tree is built starting with a single square \(S_0\) of side \(128\) cm. At every stage, two smaller squares are attached on top of each existing square along its upper edges, each tilted so that its side length is \(\dfrac{1}{\sqrt{2}}\) times the side length of the square it is attached to.

Since each square gives rise to \(2\) new squares at the next stage, the number of new squares added doubles at every stage.

 

Let \(S_0\) denote the area of the starting square.

On the basis of the information given, answer the following questions.

1. Side length of the squares added at Stage \(6\) is

 

2. Find the total area of the tree, that is, the sum of the areas of all squares from Stage \(0\) up to Stage \(5\), in terms of \(S_0\).

 

 Total area of the tree = $i{S}_0$