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If you read in the textbook pg 7 of chapter one you can find that the second formula was written to solve for frequency while the first is written to solve for energy. They are same formula written in different ways however the second one you can use once to find the frequency and then the energy difference but the first one you would have to solve for both energy levels (use the formula twice). Thus you can use either one when trying to save for the energy difference or the frequency of light you need to move between energy levels. Sorry if that was a bit confusing. The book has a better explanation on pg 7
Basically the Rydberg Formula, v=R(1/n1^2 - 1/n2^2) is derived from the previous energy formula, E=-hR/n^2. As we know, E=hv. Thus, when we subtract the formulas of energy using the energy level, E=-hR/n1^2 - (-hR/n2^2), we can plug hv to substitute for E. Then, because Planck's constant cancels out on both sides, we're just left with v=R(1/n1^2 - 1/n2^2). The reason the answer is not negative anymore is because the value of frequency cannot be negative.
You can use both equation because one of them is just the derived equation of the other. However, it is better to use the equation E=-hR/n^2 to find the energy of each "n" level then find the ∆E by subtracting the Einitial from the Efinal. From there you can find the wavelength using E=hv. Using the Rydberg equation might save you some times but it might lead to the wrong answer since you might mix up the final energy level with the initial energy level, and vice versa.
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