"If I Was a Carpenter..."
A carpenter with a 'sharp' saw decides to cut a 4"x4" down 1/3
at a time each day, forever. Assume he does this to a board 1
foot in length as follows:
1. Presuming this pattern can continue forever, fill in the table and answer the following questions.
Day What interval(s) are removed Total removed on Day 'n' Total removed after 'n' days of cutting
'n' on Day 'n'
1 (1/3,2/3) 1/3 1/3
2 (1/9, 2/9), (7/9,8/9) 1/9+1/9=2/9 1/3+2/9=5/9
3 (1/27, 2/27), (7/27, 8/27), 1/27+1/27+1/27+1/27
(19/27, 20/27), (25/27, 26/27)
4
a. Looking at the intervals that have been removed, what remains of the board as n→∞? And how
many points are remaining?
b. Write a sequence $a_n$ that gives the amount of board removed on Day 'n'.
c. Write a series that gives the total removed after 'n' days of cutting.
d. Evaluate the series in c as n→ ∞?
e. Explain the apparent contradiction between your results in (a) and (e)?
