= Solution
<Configurational entropy> counts the arrangements compatible with a macroscopic configuration: $S=k_B\ln\Omega$. Let $x=L/(Na)$, $N_\downarrow=N(1+x)/2$ and $N_\uparrow=N(1-x)/2$. The multiplicity is $N!/(N_\downarrow!N_\uparrow!)$. <Stirling's approximation> therefore gives
$$
S=Nk_B\left[\ln2-\frac{1+x}{2}\ln(1+x)-\frac{1-x}{2}\ln(1-x)\right],\qquad |x|\le1.
$$
With downward $L$ positive, the gravitational <potential energy> is $-mgL$. Minimizing the <Helmholtz free energy> $F=-mgL-TS$ gives $\operatorname{artanh}x=mga/(k_BT)$, hence
$$
\boxed{L=Na\tanh\left(\frac{mga}{k_BT}\right).}
$$
Raising <temperature> reduces the extension: the entropic preference for many up/down arrangements opposes alignment by the load. For small loads, $L\simeq Na^2mg/(k_BT)$.
For the <positive-feedback entropic chain> with replacement <force> $f(L)=f_0L/a$, $f_0>0$, the conservative potential is $U(L)=-\int_0^Lf(s)\,ds=-f_0L^2/(2a)$. Using $-f(L)L$ would overcount it by a factor two. The stationary equation becomes
$$
\boxed{x=\tanh\left(\frac{T_c}{T}x\right),\qquad T_c=\frac{Naf_0}{k_B}.}
$$
Expansion about $x=0$ gives
$$
F=F(0)+\frac{Nk_B}{2}(T-T_c)x^2+\frac{Nk_BT}{12}x^4+O(x^6).
$$
For $T>T_c$ the stable extension is zero. Below $T_c$, nonzero minima appear, with $x\simeq\pm\sqrt{3(T_c-T)/T}$ close to the transition and $|x|\to1$ as $T\to0$. Restricting to downward extensions selects the positive branch. \b[The onset is continuous in this model], with a nonanalytic mean-field change at $T_c$; a finite jump in $L$ does not follow from the printed <force> law, despite the question's discontinuity wording. A finite chain has rounded fluctuations rather than a literal thermodynamic singularity.
\Image[/past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1-chain-mean-field-transition.png]
{title=Positive-feedback chain: continuous mean-field extension and changing free-energy minima}
This is the same self-consistency equation as the zero-field mean-field treatment of the <Ising model>: the segment orientation corresponds to a spin, extension to magnetization, and the extension-dependent <force> to the self-consistent molecular field. The competition between alignment <energy> and binary configurational <entropy> explains the shared <temperature> dependence.
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