Fly shuttling between two approaching bikers

Question

Two motorcyclists are $25$ miles apart and ride straight toward each other at $20$ mph and $30$ mph. A fly flies back and forth between them at $40$ mph until they meet. What total distance does the fly cover?

Short answer

20 miles. The gap closes at $50$ mph, so they meet after $\tfrac{25}{50}=\tfrac12$ hour; flying the whole time at $40$ mph, the fly covers $20$ miles. No need to sum the infinite series of legs.

Adapted from Crack, Heard on the Street, Q1.28, p.15; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Series and the Geometric Series: Partial Sums and Convergence