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How Long Is the Cantor Set?

Age 11 to 14 Challenge Level:

In the problem The Cantor Set, we met the Cantor set, which is the limit of $C_n$ as $n$ tends to infinity.
C_1, C_2, C_3



We can talk about the length of one of our sets $C_n$.
The set $C_1$ has length 1.
The set $C_2$ has length $\frac{2}{3}$, as this is the total length of the line segments in $C_2$.

What are the lengths of $C_3$, $C_4$ and $C_5$?

Can you find a general expression for the length of $C_n$?

By considering what happens as $n$ tends to infinity, can you find the length of the Cantor set?