(Sierpinski\'s Triangle) The fractal called Sierpinski\'s triangle is the limit

Question

(Sierpinski's Triangle) The fractal called Sierpinski's triangle is the limit of a sequence of figures. Starting with an equilateral triangle with side length 1, the triangle is split into four smaller equilateral triangles and the middle triangle is removed. Then, for the remaining three equilateral triangles, each are split into four smaller equilateral triangles and the middle triangle is removed. And this process keeps going. Let Tn be the total area of removed triangles after stage n of the process. The area of an equilateral triangle with side length L is A = 3 L2/4 Find T1, and T2, the total removed after stages 1 and 2, respectively. Find a formula for Tn . Find Tn What is the area of the original triangle that remains after n rightarrowinfinity?

Explanation / Answer

Let side of original triangle be L. Whenever a triangle is removed, its side length will be half of the side length of triangle from which it is removed and remaining triangles in original body will be 3^n in number and will be equilateral with side length half. According to formula for area, its area will be 1/4 of the area of the triangle from which it is removed. Also number So T1=(3^0.5)(L^2)/16 and T2=T1+(3^0.5)(L^2)(3)/64

Tn=Tn-1 + (3^(n-0.5))(L^2)/4^(n+1)

For Lt->infinity Tn=(3^0.5)(L^2)/4 as all area of original triangle will get removed for infinite steps.

So no area of original triangle will remain after infinite steps

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