Draw a line segment of length 1, and denote it by $C_{1}$. Remove the middle third (without removing the endpoints). Denote by $C_{2}$ what remains. Now, remove the middle third (without the endpoints) of each line segment of $C_{2}$. Denote by $C_{3}$ what remains. We can continue this process, at each stage removing the middle third of each segment in $C_{n}$ to form $C_{n+1}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-04.jpg?height=193&width=721&top_left_y=294&top_left_x=672)

(a) Draw $C_{1}, C_{2}$, and $C_{3}$, indicating the numbers at the endpoints of the segments.

(b) Which of the following points belong to the Cantor set? $\frac{1}{3}, \frac{4}{9}, \frac{3}{81}, \frac{4}{81}$.

(c) What are the lengths of $C_{3}, C_{4}$, and $C_{5}$? Can you find an expression for the length of $C_{n}$?

## List 4
(a) According to the definition of the Cantor Set, we have the following drawings:

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-19.jpg?height=225&width=721&top_left_y=1161&top_left_x=672)

(b) $1 / 3$ is an endpoint of $C_{2}$, so it belongs to the Cantor Set.

$3 / 81 = 1 / 27$ and $1 / 27$ is an endpoint of $C_{4}$, so $3 / 81$ belongs to the Cantor Set.

$4 / 9$ is between $1 / 3$ and $2 / 3$, so it is in the middle third of $C_{1}$ and is removed from $C_{2}$. Therefore, $4 / 9$ does not belong to the Cantor Set.

$4 / 81$ is between $1 / 27$ and $2 / 27$, and thus is in the middle third of $C_{3}$ and is removed from $C_{4}$. Therefore, $4 / 81$ does not belong to the Cantor Set.

(c) Let's try to find a pattern for the lengths of the segments. For example, $C_{1}$ has a length of 1 and $C_{2}$ has a length of $2 / 3$. Does this already provide a pattern, where the numerator is obtained by multiplying by 2 and the denominator by 3, or in other words by $2 / 3$?

Now, $C_{3}$ has a length of $4 / 9$, $C_{4}$ has a length of $8 / 27$, and $C_{5}$ has a length of $16 / 81$. Therefore, the length of $C_{n}$ is $\left(\frac{2}{3}\right)^{n-1}$.

Note that the lengths of $C_{1}, C_{2}, C_{3}, \ldots, C_{n}, \ldots$, form a geometric progression with a common ratio $q = 2 / 3$ and the first term $a_{1} = 1$.

$$
1, \frac{2}{3},\left(\frac{2}{3}\right)^{2},\left(\frac{2}{3}\right)^{3},\left(\frac{2}{3}\right)^{4}, \ldots,\left(\frac{2}{3}\right)^{n}, \ldots
$$

## List 4