TITLE: Two trains move towards each other, a bird moves between them. How many trips can the bird make?
QUESTION [20 upvotes]: The problem statement:

Two trains move towards each other at a speed of $34\ km/h$ in the same rectilinear road. A certain bird can fly at a speed of $58\ km/h$ and starts flying from the front of one of the trains to the other, when they're $102\ km$ apart. When the bird reaches the front of the other train, it starts flying back to the first train, and so on. 


How many of these trips can the bird make before the two trains meet? 
What is the total distance the bird travels?

Commentary:
The second question of the problem seems relatively simple, since one only has to notice that the trains will take 1.5 hours to meet, therefore, the bird travels $58\cdot1.5=87 km$. However, the first question baffles me. How can one calculate how many trips the bird makes? If I'm correct, in order to obtain the time the bird will take to make its first trip, we have to add the bird's speed and the speed at which the distance of the trains is being reduced ($68\ km/h$).
This means the bird will take $\frac{102}{126}\approx0.809$ hours to finish the first trip, and the trains will be $\frac{986}{21}\approx 46.95\ km$ apart. If I continue this way (now finding how long will the bird take to travel those 46.95 km), it seems that I'll never stop or that at least it will take a huge amount of trips that cannot be computed by hand. Is there a way to find a 'quick' answer to this problem? Am I making it more complicated than it actually is? 
Thanks in advance!

REPLY [2 votes]: Just the first question: Every time the bird reaches the front of the other train, their distance has shrunk by a factor 24/92 (distance between bird and oncoming train shrinks at a rate of 92 km/h, distance between trains shrinks at a rate of 68 km/h).
After 7 trips, the bird reaches the front of the other train when the trains are about 8.4 metres apart. They crash within 0.44 seconds. There's no way the bird can stop, turn around, and reach a speed of 58 km/h again within 0.44 seconds, so the seventh trip is the last one.