3. The fractal C is constructed as follows. (i) Begin with the line segment Io = [0, 1]. (ii) Remove the open middle third (}, 3), to get I1 = [0, U3, 1]. (iii) Remove the open middle third from every remaining line segment, to get I2 (iv) Repeat the process of removing the open middle thirds of the remaining line segments infinitely many times. The first few iterations of this construction are illustrated below. (a) Find the total length of the intervals removed from [0, 1] after three iterations, i.e. the total length of the intervals removed from (0,1] to get I3. (b) By considering a suitable series, show that the total length of all of the intervals removed is equal to 1. (c) Despite the fact that an infinite number of intervals of total length 1 have been removed to construct C, there are points in the interval [0, 1] that will never be removed, i.e. there are points in C. What is one such point?
3. The fractal C is constructed as follows. (i) Begin with the line segment Io = [0, 1]. (ii) Remove the open middle third (}, 3), to get I1 = [0, U3, 1]. (iii) Remove the open middle third from every remaining line segment, to get I2 (iv) Repeat the process of removing the open middle thirds of the remaining line segments infinitely many times. The first few iterations of this construction are illustrated below. (a) Find the total length of the intervals removed from [0, 1] after three iterations, i.e. the total length of the intervals removed from (0,1] to get I3. (b) By considering a suitable series, show that the total length of all of the intervals removed is equal to 1. (c) Despite the fact that an infinite number of intervals of total length 1 have been removed to construct C, there are points in the interval [0, 1] that will never be removed, i.e. there are points in C. What is one such point?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 16E
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![3. The fractal C is constructed as follows.
(i) Begin with the line segment Io = [0,1].
(ii) Remove the open middle third (}, }), to get I = [0, ] U [G, 1).
(iii) Remove the open middle third from every remaining line segment, to get
[2 1]
I2
= [0,
(iv) Repeat the process of removing the open middle thirds of the remaining line segments
infinitely many times.
The first few iterations of this construction are illustrated below.
(a) Find the total length of the intervals removed from [0, 1] after three iterations, i.e. the
total length of the intervals removed from [0, 1] to get Iz.
(b) By considering a suitable series, show that the total length of all of the intervals removed
is equal to 1.
(c) Despite the fact that an infinite number of intervals of total length 1 have been removed
to construct C, there are points in the interval [0, 1] that will never be removed, i.e. there
are points in C. What is one such point?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4575c695-56bc-4a6a-843f-ec886ca258f2%2Fe708832c-1d69-427c-a7fd-250fd4814fe1%2Flicl5xn_processed.png&w=3840&q=75)
Transcribed Image Text:3. The fractal C is constructed as follows.
(i) Begin with the line segment Io = [0,1].
(ii) Remove the open middle third (}, }), to get I = [0, ] U [G, 1).
(iii) Remove the open middle third from every remaining line segment, to get
[2 1]
I2
= [0,
(iv) Repeat the process of removing the open middle thirds of the remaining line segments
infinitely many times.
The first few iterations of this construction are illustrated below.
(a) Find the total length of the intervals removed from [0, 1] after three iterations, i.e. the
total length of the intervals removed from [0, 1] to get Iz.
(b) By considering a suitable series, show that the total length of all of the intervals removed
is equal to 1.
(c) Despite the fact that an infinite number of intervals of total length 1 have been removed
to construct C, there are points in the interval [0, 1] that will never be removed, i.e. there
are points in C. What is one such point?
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