Really hard coupled forces problem?

Question

Imagine a train consisting of N frictionless cars following a locomotive that is accelerating the whole train forward with acceleration of magnitude a. Assume that the first car behind the locomotive has mass M. If the tension in the coupling at the rear of each car is 10% smaller than the tension in the coupling in the front of the car, what is the mass of each sucessive car as a fraction of M? What is the total mass of the cars in terms of M? [Hint: You might find it helpful to know that 1+x^2+x^3+...+x^n=(1-x^n+1)/(1-x).]

Answer 1

F=Ma

Ft= F + .9F + .81F +...F(.9)^N) where N - number of cars +1

Ft=F(1+.9 + .81+...+(.9)^N) it is geometric series (see reference below) we have

Ft=F(1- (.9)^(N+1) / (1-(.9))

Let's test
For N=1 (2 cars)
Ft=Ma(1.9)

For N=10 (11cars)
Ft= Ma(9.69)

For N=100
Ft= Ma(9.9999 )

For N-> infinity
Ft= Ma(1/.1)=10 Ma

Mt= Ft/a= 10M