homework and exercises - Infinite Atwood's machine

I tried doing problem 43 in this collection just for fun. It is reproduced as follows:

(I'm not allowed to post the entire problem here since I'm a new user.)

Here are my equations:
$$T - mg = ma$$ (equation 1)
$$Mg - T = Ma$$ (equation 2)
(Upper case "M" used for infinite series of masses on the right arm of the first pulley. Lower case "m" for the small, single mass on the left arm.)

algebra is as follows:
$$Mg - (ma + mg) = Ma$$
$$g (M - m) = a (M + m)$$
$$a = g (M - m) / (M + m)$$

Here is my reasoning to get final answer:

The limit of $a$ for large $M$ goes to $g$. Therefore, acceleration of the top mass is $g$. However, the correct answer is $g/2$. Why does my reasoning get the wrong answer? Any ideas?

(Please excuse the formatting, I'm a newbie here).

Answer

Your second equation is wrong. You can't consider a nonrigid system as a single blob of mass $M=\Sigma m$. IIRC, this problem is done via a recursive relation.