An infinite atwood machine Below is a seemingly simple problem involving an infinite system of pulleys.

Consider the infinite Atwood’s machine shown below. A string passes
  over each pulley, with one end attached to a mass and the other end
  attached to another pulley. All the masses are equal to $m$, and all
  the pulleys and strings are massless. The masses are held fixed and
  then simultaneously released. What is the acceleration of the top
  mass?

(We’ll define this infinite system as follows. Consider it to be made
  of $N$ pulleys, with a non-zero mass replacing what would have been the
  $(N + 1)$st pulley. Then take the limit as $N\to\infty$.)

 A: First of all, the question is delightful. I did look at the solution but I didn't understand it too. So I came up with this one (maybe it resembles the standard solution, maybe not) . 
See, if we create a similar pulley system in a smart way such that it would behave exactly the same like this one and see it working then we can easily get an answer. 
To make such a system, you must agree that $$\infty + 1 = \infty$$ 
So if you exclude the first pulley, what remains is the same system of pulleys because if you take away one pulley from the system, that wont make a difference because there are an infinite number of them. This is what our  system will look like -

Now, what if we say that let us suppose that the tension in string $a$ is $mg$ . What will be its consequences? 
Well, the first ball won’t move because on it the forces will be balanced. 
Because of this, the second pulley won’t move! Due to this our system will completely behave as the system which is given in the question because the top pulley in the question is not able to move and the top pulley in our system will also not move.
As the pulley’s are massless, the tension in $b$ will be $\frac{mg}{2}$ . Hence the net force on the first ball will be $ mg - \frac{mg}{2}   = \frac{mg}{2}$ which will produce an acceleration of $\frac{g}{2}$ in the first ball of our system which corresponds to the first ball of the system given in the question and as a matter of fact, the first ball of the system given to us in the question will also move with an acceleration go $\frac{g}{2}$ !
