Please tell me how to solve this? It's homework due, dead urgent as well!?

Question

TIme to help me, now, it's very urgent, and I'm stuck:

Question 2:

This is a Michelson-Morley Interferometer (Diagram not part of the question):

http://www.physics.sfsu.edu/%7Errogers/Phys%20321/Image28.gif

Question: A glass container is placed along arm 2 (the arm right above L_A in the diagram) of the interferometer. The glass container is 1 cm long (you may ignore the thickness of the glass).

At first, the container is empty and the mirror along the horizontal line (the mirror that is placed vertically) is adjusted so that a bright fringe is observed at the detector. Now, a gas of unknown index of reflection is slowly allowed to enter to container. During the time this occurs, 1624 changes between bright and dark are observed to occur while it is filling up. The source is known to have a wavelength of 550 nm. What is the index of refraction for this gas?

Answer 1

If the wavelength is 550nm in the air, in the gas it should be shorter.

So everytime you see a change betwenn black and white is like the wave compresses inside the glass container.

In 1cm of gas there are 1cm / 550 nm = 18182 wave fronts

When the gas container is filled there are 18182+1624 wave fronts inside

So 1cm /19806 = 505 nm (wavelength inside the gas chamber)

Index of refraction is 550/505 = 1.09

Keep in mind that I made some approximations with decimals.