i was googling the similar question earlier, hoping that i would find some information on this topic (perturbation), and i found post, i suppose you are taking CHE320 and this is one of the questions in practice problem right?

have you figured out the answer or an explanation for this? for n=6 i kinda agree with your explanation (wavefunction is at a node, probability=0), but the question says the wavelength is opposite to n=5..

and for n=5, the center is at the top of a peak, i guess we are wondering the same question, would the decrease in potential energy increase or decrease the gap...?