Few days ago, I was asked the following question:
There are $2$ cities. city $A$ and city $B$ with distance $d$=600km
There are $2$ trains with speed of $vt$ = 100km/h.
There is $1$ fly with speed of $vf$ = 300km/h.
Train 1 goes from city A to city B , while Train 2 goes in the opposite direction.(both start at the same time). the fly starts from train 1 and go to train 2 , than back to train 1 , than back to train 2...etc...
What is the distance that the fly passes till the trains cross each other ?
The answer is simple - it takes 3 hours for the train to cross -> fly was running 900km($300 \times 3$).
I was wondering on the "hard" way of solving this problem by actually calculating the distance the fly pass on each stage. It appears to be the sum of a (finite?) series of the distances.
Could you help me solve this problem the "hard way" ?