Question

Find the three longest wavelengths (call them λ1, λ2, and λ3) that "fit" on the string, that is, those that satisfy the boundary conditions at x=0 and x=l. these longest wavelengths have the lowest frequencies. express the three wavelengths in terms of l. list them in decreasing order of length, separated by commas.

Answer 1

If you have a string that is fixed on both ends the amplitude of the oscillation must be zero at the beginning and the end of the string. Take a look at the pictures I have attached. It is clear that  our fundamental harmonic will have the wavelength of:
$\lambda_1=\frac{L}{2}$
All the higher harmonics are just multiples of the fundamental:
$\lambda_n=n\lambda_1\\ \lambda_n=n\frac{L}{2}$
Three longest wavelengths are:
$n=3; \lambda_3=\frac{3L}{2}\\ n=2; \lambda_2=L\\ n=1; \lambda_=\frac{L}{2}$