Double-Slit Interference and Diffraction Gratings

Interference depends on the path length difference measured in wavelengths: whole numbers give constructive interference and odd half-integers give destructive, so the raw difference in metres settles nothing until it is divided by $\lambda$. For two slits separated by $d$, $d\sin\theta = m\lambda$ locates the maxima with the central one at $m = 0$, and the linear form $y = m\lambda L/d$ follows only while the angle is small. A grating obeys the same condition, so more slits leave the maxima where they were and make them far narrower, and because $\theta$ rises with $\lambda$ the red end of each order sits farthest from the centre.

Six errors dominate. Deciding constructive or destructive from the raw path difference rather than from its ratio to the wavelength. Using $y = m\lambda L/d$ at angles where the small-angle approximation has already failed, which gratings routinely produce. Counting the central bright fringe as the first order, which shifts every position by a fixed ratio. Swapping the slit separation with the screen distance, or reading $d$ as the width of one slit. Treating a grating as a double slit with extra openings, so adding slits is expected to smear the pattern. And putting violet at the outside of a grating spectrum because a prism bends violet the most.