 Solution

Hydrogen's $n=2$ excitation energy is $3R/4=10.2$ eV. With electronic degeneracy ratio $g_2/g_1=4$, its relative population is $4e^{-10.2/1.60}\simeq 6.8\times 10^{-3}$. Thus the first excited manifold contributes below one percent of the ground population at the inferred temperature, explaining the leading ground-state approximation.

There is a qualification to extending this argument to infinitely many Rydberg states: the isolated-atom partition sum with degeneracy $2n^2$ formally diverges. Plasma screening and level dissolution impose a physical cutoff. The small low-lying populations justify the simple approximation in its intended regime, not an unrestricted infinite partition sum.