Working with wave number equations, use trial-and-error?
[&hellip;] my teacher told me to continue using a trial and error method [&hellip;]


But nobody told you to do it manually once you have understood how to adjust the Rydberg constant for $\ce{He+}$ and use the formula to get the energies for the given  transistions, in order to compare them with the $\ce{H_{\alpha}}$ line (656.3 nm):

import scipy.constants

transitions = ((7,5), (5,3), (6,4), (4,2))

def get_he_line_nm(n_upper, n_lower):
    '''calculate the energy for a line in a He+ spectrum and return 
       it in nanometer''' 

    energy = 4 * scipy.constants.Rydberg * (pow(n_lower, -2) - pow(n_upper, -2))

    return 1.0/(scipy.constants.nano * energy) 
    # energy in nm


for upper, lower  in transitions:
    wavelength = get_he_line_nm(upper, lower)
    print('{0} -&gt; {1} : {2:7.2f} nm'.format(upper, lower, wavelength)) 


will give the output


7 -&gt; 5 : 1162.81 nm
5 -&gt; 3 :  320.37 nm
6 -&gt; 4 :  656.11 nm
4 -&gt; 2 :  121.50 nm