The fractal called the Sierpinski triangle is the limit of a sequence of figures. Starting with the

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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 1/2 is removed. Then, three inverted equilateral triangles with sides of length 1/4 are removed from this figure (see figure). The process continues in this way. Let Tn 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 = √3L2/4.

a. Find T1 and T2, the total area of the removed triangles after stages 1 and 2, respectively. 

b. Find Tn, for n = 1, 2, 3,  . . . .
c. Find lim Tn. п500

d. What is the area of the original triangle that remains as n →∞?

1 Second stage Initial stage First stage

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Calculus Early Transcendentals

ISBN: 978-0321947345

2nd edition

Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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