[s3a]

I did question number 4 and the answer is λ = 434 nm but I get λ = -434nm. I noticed that if I were to switch the n=5 and n =2 and have n=5 be n2 and n=2 be n1 then I would get the same exact answer but the question says FROM n = 5 state TO the n = 2 state. So, essentially, I am having a little bit of trouble understanding the words so this is a "word problem" more than a problem of knowing which formula to apply or something of the sort.

(My work is attached as a pdf file)

Any input would be greatly appreciated!
Thanks in advance!

[renge ishyo]

The sign depends on how you are plugging the n values into the rydberg equation. If you plug in the n values as in the equation below, the sign on the wavelength will always come out positive:

1/λ = R(1/n_low² - 1/n_high²)

If you switch the order of the n values inside the parenthesis then the wavelength will come out negative. It's not wrong per say, it would just mean that your energy was negative (corresponding to emission of light). If you are concerned about this you can just use the above formula to always get positive energies and wavelengths. Then if you deduce from the question that the process is emission you can throw a minus sign onto your energy as a final step.

[s3a]

So that means the way I did it is correct? To make somewhat of a comparison, this is like saying -100 kJ which means releasing 100 kg, right? In other words, the negative sign symbolizes some kind of direction as opposed to a lack of something, right?

[renge ishyo]

You did solve the problem correctly. An atom transitioning from n=5 to n=2 would release a photon in the process, which would correspond to a negative energy. If you then solve the negative energy for the wavelength you will unfortunately get a negative wavelength. Since the negative sign just means the direction you can simply drop it and report a positive wavelength if you'd like.