There is a puzzle which says:
Two kids on bikes are facing each other, 20km apart on a perfectly straight road. There is a fly on the handlebars of one bike.
The kids begin to pedal towards each other at 10km/h, and the fly sets off at 15km/h, at the same moment. When the fly meets the other bike, it turns around and flies back, and so on.
The kids and the fly keep a steady pace without rests until they meet, when they stop.
How far does the fly fly?
The puzzle looks formidably difficult, but the trick is that there's a simple solution: the kids took an hour to meet, during which time the fly flew at 15km/h. Thus the fly flew 15km.
What I'm wondering is how you'd go about solving the problem without that trick. Given the iteration up to a limit, would it be some sort of integral calculus question?
It’s a geometric series. You can figure out how far the fly flies to meet the other bike for the first time. At that instant the bikes are much closer than at the start. The ratio of that distance to the original distance is the ratio of the series.
There is a famous story about John von Neumann in which it is said someone asked him a question similar to this. He answered so quickly they said he must have known the trick. But he claimed to have summed the series.