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Question 5 The figure below shows a fractal pattern emerging from a circle inscribed in an equilateral triangle with side lengths 1. On each step, 3 circles are added to the previous image as demonstrated. Due to the nature of this construction, we can repeat this process indefinitely. Let An be the area of a circle constructed on the nth iteration (with A1 being the original inscribed circle). Let Sn be the total area of all the circles at the nth iteration. (a) Explain why Sn must converge. (b) Write an expression for Sn in terms of An. (c) Calculate A1 given that the original circle has radius 1/(2sqrt(3)). (d) By considering the ratio between A2 and A1, find an expression for An. (e) Calculate the value of lim n->infinity Sn.

          Question 5
The figure below shows a fractal pattern emerging from a circle inscribed in an equilateral triangle with side lengths 1. On each step, 3 circles are added to the previous image as demonstrated. Due to the nature of this construction, we can repeat this process indefinitely.
Let An be the area of a circle constructed on the nth iteration (with A1 being the original inscribed circle). Let Sn be the total area of all the circles at the nth iteration.
(a) Explain why Sn must converge.
(b) Write an expression for Sn in terms of An.
(c) Calculate A1 given that the original circle has radius 1/(2sqrt(3)).
(d) By considering the ratio between A2 and A1, find an expression for An.
(e) Calculate the value of lim n->infinity Sn.

$Question 5 The figure below shows a fractal pattern emerging from a circle inscribed in an equilateral triangle with side lengths 1. On each step, 3 circles are added to the previous image as demonstrated. Due to the nature of this construction, we can repeat this process indefinitely. Let An be the area of a circle constructed on the nth iteration (with A1 being the original inscribed circle). Let Sn be the total area of all the circles at the nth iteration. (a) Explain why Sn must converge. (b) Write an expression for Sn in terms of An. (c) Calculate A1 given that the original circle has radius 1/(2sqrt(3)). (d) By considering the ratio between A2 and A1, find an expression for An. (e) Calculate the value of lim n->infinity Sn.$

Added by David H.

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