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Saga of the Indecisive Insect

Bill the Bug is about to travel along the number line.

He starts at 0.

Then he crawls half-way to 3.

Then he crawls half-way back to 0.

Then he crawls half-way back to 3.

Then he crawls half-way back to 0.

Then he crawls half-way back to 3.

Then he crawls half-way back to 0.

Then he crawls half-way back to 3.

Then he crawls half-way 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 right-hand turning points (RHTP) are 3/2, 15/8, 63/32,...

The left-hand 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 = (2n+1-1)/(2n) and LHTP = (2n+1-1)/(2n+1).

But, these can be simplified to RHTP = 2 - 1/(2n) and LHTP = 1 - 1/(2n+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).