Question

How do you know when to use which form of Rydberg's constant in his equation when looking for n values?

Answer 1

There is only one Rydberg's constant, and that is
\[R=1.097*10^{7}m^{-1}\]

If you went through the derivation for hydrogen-like atoms, then you will find that the rydberg's constant becomes:
\[\frac{Z^{2}e^{4}m_{e}}{8\epsilon_{0}^{2}h^{3}c}=Z^{2}R\] or something like that.

Then the Rydberg formula simply becomes
\[\frac{1}{\lambda}=Z^{2}R(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}})\]

Answer 2

Where Z is the number of protons in the atom. Forgot to mention that :P