= Solution
For the <uniform-binding multilayer adsorption model>, $\theta_j$ describes sites of \b[exactly] height $j$, not the fraction whose height is at least $j$. At equilibrium the upward and downward fluxes on each adjacent pair of heights balance:
$$
k_a p\theta_{j-1}=k_d\theta_j,\qquad j\geq1.
$$
For completeness, stationarity alone gives a height-independent probability current; the reflecting boundary at height zero sets that current to zero. This justifies pairwise balance for a normalizable equilibrium distribution.
Writing $x=K_{\rm ads}p$ gives the recurrence $\theta_j=x\theta_{j-1}$. Successive substitution then yields
$$
\theta_1=x\theta_0,\qquad \theta_2=x^2\theta_0,\qquad
\boxed{\theta_j=x^j\theta_0=(K_{\rm ads}p)^j\theta_0}.
$$
An equilibrium <geometric distribution> of unbounded heights requires $0\leq x<1$. The first-layer-only <Langmuir adsorption model> instead truncates the allowed heights at one, which changes the normalization and removes this convergence restriction.
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