Michelson interferometer moves and bright fringes disappear

[Cocoleia]

Homework Statement

In a Michelson interferometer, as one of the mirrors is moved, 250 bright fringes disappear into the centre. If the light wavelength was 800nm, how far was the mirror moved?

Homework Equations

The Attempt at a Solution

My logic is:
If we were to count how many passing bright fringes we have, we could find the wavelength, basing ourselves off of the formula: 2d=mλ. In this case I would say 2Δd=Δmλ, where Δd is the distance that the mirror moved and Δm is the number of passing bright fringes.
So I would solve for Δd and get 0.01cm
My problem is, I am pretty sure I need to somehow derive 2d=mλ, I can't just use it. Can someone get me started on how to derive this?
Also, is my logic correct

 

[BvU]

What happens to the corresponding optical path length if a mirror is translated over $\lambda\over 2$ ? Idem ${2\lambda\over 2}, {3\lambda\over 2}$, etc ?