Fly shuttling between cars 400 miles apart

Two cars sit at the two ends of a straight $400$-mile road and set off toward one another at the same moment: car $A$ moves at $120$ mph and car $B$ at $60$ mph. At the start, a fly leaves car $B$ heading for car $A$ at $180$ mph. Each time it reaches one car it turns around instantly and heads for the other, keeping the same speed, and it carries on doing this until the two cars collide (it never escapes). What total distance does the fly cover?

$400$ miles. The cars close the gap at $120+60=180$ mph, so they meet after $400/180=\tfrac{20}{9}$ hours. The fly flies at $180$ mph for that whole time, covering $180\cdot\tfrac{20}{9}=400$ miles; there is no need to sum the infinitely many legs.