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