Chapter 9, Problem 11PS

To determine

Write a formula for the area of the triangles generated in the $nth$ stage

Answer to Problem 11PS

  $A_n=\frac{\sqrt{3}}{4}\times {\left({\left(\frac{1}{2}\right)}^{n-1}\right)}^2$

Explanation of Solution

Calculation:

The sierpinski triangle is shown below

  

Length of each side of the original triangle is one unit

Mid-point of each side is considered and another triangle is drawn, dividing the original triangle into four triangles

In this case each side of the triangle

  $=\frac{1}{2}Units$ or $0.5units$

If this exercise is carried out once more, the side of each triangle

  $=\frac{1}{4}Units$ or $0.25units$

When the exercise is carried out once more, the side of each triangle

  $=\frac{1}{8}Units$ or $0.125units$

Thus side length of the triangle that will generated in the $nth$ stage is $S_n={\left(\frac{1}{2}\right)}^{n-1}$

Each side of the triangle is of one unit

It is an equilateral triangle and area of an equilateral triangle is $=\frac{\sqrt{3}}{4}\times s^2$

Where $s$ is the length of each side of the equilateral triangle

Hence, according to (1), side length of triangles that will be generated in the $nth$ stage is

  $A_n=\frac{\sqrt{3}}{4}\times S_n{}^2$ Or

  $A_n=\frac{\sqrt{3}}{4}\times {\left({\left(\frac{1}{2}\right)}^{n-1}\right)}^2$