Saga of the Indecisive Insect
Bill the Bug is about to travel along the number line.
He starts at 0.
Then he crawls halfway to 3.
Then he crawls halfway back to 0.
Then he crawls halfway back to 3.
Then he crawls halfway back to 0.
Then he crawls halfway back to 3.
Then he crawls halfway back to 0.
Then he crawls halfway back to 3.
Then he crawls halfway back to 0.
....etc.
Using mathematics, describe Bill the Bug's travels...
 His stoppping or turning points?
 Where he is likely to end up?
Hint: Make a chart (a spreadsheet works great) of Bill's travels by finding his stopping or turning points (e.g. 0, 3/2, 3/4, 15/8, ...).
Also, to see the pattern, it is best to use fraction notation.
Solution Commentary: The righthand turning points (RHTP) are 3/2, 15/8, 63/32,...
The lefthand turning points (LHTP) are 3/4, 15/16, 63/64,...
Thus, LHTP = 1/2 of the previous RHTP.
And, we can represent the two respective turning points by RHTP = (2^{n+1}1)/(2^{n}) and LHTP = (2^{n+1}1)/(2^{n+1}).
But, these can be simplified to RHTP = 2  1/(2^{n}) and LHTP = 1  1/(2^{n+1}).
Thus, the RHTP approaches 2 from the left and the LHTP approaches 1 from the left as well.
As n increases without bound, Bill the Bug is essentially alternating between the points 1 and 2...though never reaches either as a stopping point (but he does pass through 1 an infinite number of times during his travels).
