Problem 9
Assume you have a strip of paper of a
very precise length (one meter) and very precise measuring and paper
cutting instruments (more precise than anything that is known to
us!). You start to perform operations on this strip of paper in the
following pattern:
Step one. Split the original strip into
thirds then cut out the middle third. Throw it away.
(You are left with two strips of paper
that each have a length of one third of a meter.)
Step two. Split the strips you have
left into thirds then cut out the middle third of each strip and
throw it away.
You are left with four strips of paper
that each have a length of one ninth of a meter.)
This pattern is repeated over and over
again. Generally:
Step ‘
n’. Split the
strips you have left from the previous step into thirds then cut out
the middle third of each strip and throw it away.
Assume that you can perform these steps
infinitely many times. Your task is now to sum up the lengths
of paper that you throw away at each step i.e. to find the total
length of paper that you would throw away if you performed infinitely
many steps.
[HINT: Draw a picture of the process
that you are carrying out (either on paper or in your head). Find a
pattern in the lengths of strips that you throw away at each step.
Using this pattern, make an infinite sum for the lengths you throw
away. Is this sum familiar to you? Do you remember the formula for
its value?]
