Von Neumann, even cleverer than that

There is an old puzzle which goes "Two cyclists start out 50 miles apart, and head towards each other, each one going 10 mph. At the same instant, a fly leaves the first bike and flies at 30 mph towards the second. When it gets there, it immediately turns around and heads back to the first. Then it repeats, going back and forth between the two bikers. By the time they reach each other, how far will the fly have travelled?"

( Spoilers, )