= Solution
At room temperature, $k_BT$ is of order $0.026\,\mathrm{eV}$, far smaller than the electronic excitation gaps of the inferred hydrogenic ion. Ground-state absorption is therefore the natural interpretation. Assign the three lines to $1\to2$, $1\to3$, and $1\to4$. Dividing each reciprocal wavelength by its corresponding factor $1-1/n^2$ gives
$$
R_{\rm eff}=R_MZ^2\simeq(9.8765,9.8684,9.8765)\times10^7\,\mathrm{m^{-1}}.
$$
The small scatter is consistent with the rounded wavelength data. Since $R_\infty\simeq1.0974\times10^7\,\mathrm{m^{-1}}$ and $\mu\simeq m_e$,
$$
\boxed{R_{\rm eff}\simeq9.87\times10^7\,\mathrm{m^{-1}},\qquad Z\simeq\sqrt{R_{\rm eff}/R_\infty}=3.}
$$
The ion is thus hydrogen-like lithium, $\mathrm{Li}^{2+}$. If “effective <Rydberg constant>” denotes only the reduced-mass constant rather than the charge-inclusive spectral coefficient, its value is $R_M\simeq R_{\rm eff}/9\simeq1.097\times10^7\,\mathrm{m^{-1}}$. The two conventions are explicitly distinguished; the rounded lines do not determine a precise isotope mass.
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