More puzzles

Dear Diary,

John von Neumann was a mathematician alive when quantum mechanics was getting big.  Allegedly, he was faster and smarter than Bethe.   Somehow I can't quite believe that...Bethe was used to calculate mental math to check the first computers.

But anyway, there's a story about von Neumann and a puzzle:  Given two bikes twenty miles apart and on a course to collide (each bike travels at 10 mph and starts 20 miles apart), a fly travels from one tire's edge to the tire on the other bike repeatedly until is it squashed (always at 15 mph--it turns around instantaneously on reaching the bike).

How far does the fly travel?

There's an easy way to get this answer and it's to recognize that it takes the bikes 1 hour to meet, so the fly travels 15 miles.

But there's a hard way, and apparently von Neumann calculated this instantly on hearing the question.  Once answered, the questioner said, "Oh, you must know the trick." (solution presented above) But von Neumann said, "What trick?  I just summed up the geometric series."

Oh man, diary!  How do you even set up the geometric series for this?  I'd love to know!