Draw a line segment with a unit length, denoting it by $C_{1}$. Remove the central third (without removing the endpoints) and denote by $C_{2}$ what remains. Now, remove the central third (without the endpoints) of each of the two line segments that make up $C_{2}$, denoting by $C_{3}$ what remains. This process can be continued, always removing, at each stage, the central third of each segment in $C_{n}$ to form $C_{n+1}$. The Cantor set is formed by the elements of $C_{1}$ that are never removed, at any stage.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-031.jpg?height=200&width=714&top_left_y=1171&top_left_x=685)

(a) In the given figure, indicate the numbers at the endpoints of the segments $C_{1}, C_{2}$, and $C_{3}$.

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

(c) What are the lengths of $C_{3}, C_{4}$, and $C_{5}$? Can you find an expression for the length of $C_{n}$?
(a) According to the definition of the Cantor set, we have the following drawing.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-106.jpg?height=232&width=720&top_left_y=1072&top_left_x=725)

(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 $C_{1}$, which was removed from $C_{2}$; thus, $4/9$ does not belong to the Cantor set. $4/81$ is between $1/27$ and $2 / 27$, so it is in the middle third of the first segment of $C_{3}$, which was removed from $C_{4}$; thus, $4/81$ does not belong to the Cantor set.

(c) Observe that $C_{1}$ has length $1, C_{2}$ has length $2 / 3, C_{3}$ has length $4 / 9$, $C_{4}$ has length $8 / 27$ and $C_{5}$ has length 16/81. Thus, the lengths of $C_{1}, C_{2}, C_{3}, \ldots, C_{n}, \ldots$ form a geometric progression with ratio $q=2 / 3$ and first term $a_{1}=1$, as follows.

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

In particular, the length of $C_{n}$ is $(2 / 3)^{n-1}$.