Question 43
Singer Set - Draw a line segment of length $1$, and denote it by $C_{1}$. Remove the central third (without removing the extremes). Denote by $C_{2}$ what was left. Now, remove the central third (without the extremes) of each line segment of $C_{2}$. Denote by $C_{3}$ what was left. We can continue this process, at each stage by removing the central third of each segment in $C_{n}$ to form $C_{n+1}$. 
a. Draw $C_{1}, C_{2}$ and $C_{3}$, indicating the numbers at the ends of the segments.
b. Which of the following points belong to the singer 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 $C_{n}$?
a. Draw $C_{1}, C_{2}$ and $C_{3}$, indicating the numbers at the ends of the segments.
b. Which of the following points belong to the singer 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 $C_{n}$?