Leaving Cert Mathematics Sequences & Series: The closed line segment [0, 1] i

Question

The closed line segment [0, 1] is shown below. The first three steps in the construction of the Cantor Set are also shown:

- Step 1 removes the open middle third of the line segment [0, 1], leaving two closed line segments (i.e. the end points of the segments remain in the Cantor Set)
- Step 2 removes the middle third of the two remaining segments leaving four closed line segments
- Step 3 removes the middle third of the four remaining segments leaving eight closed line segments.

The process continues indefinitely. The set of points in the line segment [0, 1] that are not removed during the process is the Cantor Set.

(a) (i) Complete the table below to show the length of the line segment(s) removed at each step for the first 5 steps. Give your answers as fractions.

Step   | 1 | 2 | 3 | 4 | 5 |
Length Removed |   |   |   |   |   |

(ii) Find the total length of all of the line segments removed from the initial line segment of length 1 unit, after a finite number (n) of steps in the process. Give your answer in terms of n.

(iii) Find the total length removed, from the initial line segment, after an infinite number of steps of the process.

(b) Complete the table below to identify the end-points labelled in the diagram. Give your answers as fractions.

Label   | A | B | C | D | E | F |
End-point |   |   |   |   |   |   |

(ii) Give a reason why  $rac{1}{3} + rac{1}{27} + rac{1}{81}$ is a point in the Cantor Set.

(iii) The limit of the series $rac{1}{3} + rac{1}{27} + rac{1}{243} + ullet ullet ullet$ is a point in the Cantor Set. Find this point.

The closed line segment [0, 1] is shown below - Leaving Cert Mathematics - Question 7 - 2019

Worked Solution & Example Answer:The closed line segment [0, 1] is shown below - Leaving Cert Mathematics - Question 7 - 2019

Complete the table below to show the length of the line segment(s) removed at each step for the first 5 steps.

Find the total length of all of the line segments removed from the initial line segment of length 1 unit, after a finite number (n) of steps in the process.

Find the total length removed, from the initial line segment, after an infinite number of steps of the process.

Complete the table below to identify the end-points labelled in the diagram.

Give a reason why $rac{1}{3} + rac{1}{27} + rac{1}{81}$ is a point in the Cantor Set.

The limit of the series $rac{1}{3} + rac{1}{27} + rac{1}{243} + ullet ullet ullet$ is a point in the Cantor Set. Find this point.

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