Explanation
Rearranged RydbergBohr Equation to Find ni or nf
The RydbergBohr equation is applicable for calculating the wavelengths of spectral lines in hydrogenlike atoms. To find the initial ($n_i$) or final ($n_f$) principal quantum numbers, the equation can be rearranged. The general form of the Rydberg formula for the wavelength $\lambda$ is:
$\frac{1}{\lambda} = R \left( \frac{1}{n_f^2}  \frac{1}{n_i^2} \right)$where $R$ is the Rydberg constant. To isolate $n_i$ or $n_f$, we follow these steps:

Isolating $\frac{1}{n_i^2}$ or $\frac{1}{n_f^2}$:
Now, rearrange it to solve for the desired term:
$\frac{1}{n_i^2} = \frac{1}{n_f^2}  \frac{1}{\lambda R}$or
$\frac{1}{n_f^2} = \frac{1}{n_i^2} + \frac{1}{\lambda R}$
Solving for $n_i$:
To find $n_i$, use the isolated term for $\frac{1}{n_i^2}$:
$\frac{1}{n_i^2} = \frac{1}{n_f^2}  \frac{1}{\lambda R}$Then take the reciprocal to find $n_i^2$:
$n_i^2 = \frac{1}{\left( \frac{1}{n_f^2}  \frac{1}{\lambda R} \right)}$Finally, take the square root to find $n_i$:
$n_i = \sqrt{\frac{1}{\left( \frac{1}{n_f^2}  \frac{1}{\lambda R} \right)}}$
Solving for $n_f$:
To find $n_f$, use the isolated term for $\frac{1}{n_f^2}$:
$\frac{1}{n_f^2} = \frac{1}{n_i^2} + \frac{1}{\lambda R}$Then take the reciprocal to find $n_f^2$:
$n_f^2 = \frac{1}{\left( \frac{1}{n_i^2} + \frac{1}{\lambda R} \right)}$Finally, take the square root to find $n_f$:
$n_f = \sqrt{\frac{1}{\left( \frac{1}{n_i^2} + \frac{1}{\lambda R} \right)}}$
Summary
 For $n_i$:
 For $n_f$:
Use these rearranged equations to determine the initial or final principal quantum numbers in atomic transitions.