Cantor Middle Thirds Set

Consider the Cantor set as pictured in Gleick, p. 93. This set is constructed as follows. Initially, at step $0$, start with a line segment of length $1$. At each successive step, remove the middle third of each line segment in the previous step.

Complete the following table.

Step	Number of Pieces	Length of Each Piece	Total Length in Set
$0$	$1$	$1$	$1$
$1$
$2$
$3$
$4$
$n$

1. As $n$ gets larger and larger, what happens to the number of pieces in the set?
   

2. As $n$ gets larger and larger, what happens to the length of each piece in the set?
   

3. As $n$ gets larger and larger, what happens to the total length of the total set?