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Consider two trains moving in opposite directions on the same track. The trains start simultaneously from two towns, Aville and Bville, separated by a distance d. Each train travels toward each other with constant speed v. A bee is initially located in front of the train in Aville. As the train departs Aville, the bee travels with speed u>v along the track towards Bville. When it encounters the second train, it instantaneously reverses direction until it encounters the first train, then it reverses again, etc. The bee continues flying between the two trains until it is crushed between the trains impacting each other. The purpose of this problem is to compute the total distance flown by the bee until it is crushed. Assume that the bee is faster than the trains.

Find an expression for the distance d_n covered by the bee after its nth encounter with a train. Define d_0 as the distance traveled during the first flight from Aville towards the train near Bville, d_1 the distance traveled by the bee during the first trip from the Bville train to the Aville train, etc. Sum the resulting series to get the final answer.

I don't know how to proceed with this exercise. Could someone help me please?

Thank you in advance

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# Homework Help: Two Trains and a Bee

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