Let's see.....
The bicyclists will take one hour to reach each other at the midway point, 10 miles distant from where each started. Since the fly travels continuously in this time....he must travel 15 miles in the one hour.
Seen another way....
The total distance the fly travels in meeting the first distant bike = 12 miles. The successive distances traveled between the converging bikes is just (1/5) of the preceding distance. To see this.....note that the distant bike and fly are 20 miles apart when they begin. They close 2.5 miles of distance between them every 6 minutes. So...they meet in (20/2.5) x 6 = 48 min. And in this time, the fly has traveled 4/5 hr x 15 mph = 12 miles.
So. the bikes have traveled 16 miles total in this time and are 4 miles apart. It takes the fly and the first bike (4/2.5) x 6 = 9.6 minutes to close this distance. And in this time, the fly travels (9.6/60) x 15 = 2.4 miles. Note that this is 1/5 of the previous distance it has traveled = 12 miles. And this pattern continues.
So....summing the series, we have :
12 + 12/5 + 12/25 + 12/125 + .... = 12 /(1 - 1/5) = 15 miles
