Sierpinski triangle The fractal called the Sierpinski
Chapter 1, Problem 59RE(choose chapter or problem)
Sierpinski triangle The fractal called the Sierpinski triangle is the limit of a sequence of figures. Starting with the equilateral triangle with sides of length 1, an inverted equilateral triangle with sides of length \(\frac{1}{2}\) is removed. Then, three inverted equilateral triangles with sides of length \(\frac{1}{4}\) are removed from this figure (see figure). The process continues in this way. Let \(T_{n}\), be the total area of the removed triangles after stage n of the process. The area of an equilateral triangle with side length L is \(A=\sqrt{3} L^{2} / 4\).
a. Find \(T_{1}\), and \(T_{2}\), the total area of the removed triangles after stages 1 and 2, respectively.
b. Find \(T_{n}\), for n = 1, 2, 3,....
c. Find \(\lim \limits_{n \rightarrow \infty} T_{n}\).
d. What is the area of the original triangle that remains as \(n \longrightarrow \infty\)?
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