Q.Derive lens makers' equation. Why is it called so? Under which conditions focal length f and radii of curvature R are numerically equal for a lens?

[!TLDR] Applying refraction at a spherical surface twice (once per lens surface) and adding gives 1/f=(n-1)(1/R1-1/R2), called the lens maker's equation because it tells the optician which radii to grind; f=R numerically when the lens is symmetric double-convex/concave with n=1.5. [!ANSWER] 1/f = (n-1)(1/R1 - 1/R2); name …

Consider a thin lens of radii R1 (first surface, facing the object) and R2 (second surface), made of material index n, in a surrounding medium of index 1, with the lens thin enough that both surfaces share a common pole P. For an object O at distance u from P, applying the single-spherical-surface formula to the FIRST surface (with n1=1, n2=n, R=R1) gives an equation for an intermediate image at v1. The ray, before actually reaching this intermediate image, is intercepted by the SECOND surface, for which the intermediate image acts as a (virtual) object; applying the single-surface formula again to this SECOND surface (now with n1=n, n2=1, R=R2, object distance -v1) gives a second equation, this time for the true final image at v. ADDING these two equations together makes the intermediate distance v1 cancel out exactly, leaving 1/f = (n-1)(1/R1 - 1/R2) -- the LENS MAKER'S EQUATION. It is called this because it is precisely the equation an optician (lens-maker) uses in reverse: given a desired focal length f (e.g. to correct a particular spectacle power) and a chosen glass of index n, it tells them exactly which two radii of curvature R1 and R2 to actually grind into the glass. …

Derive by applying the single-spherical-surface formula successively to each of the lens's two surfaces and adding the results to eliminate the intermediate image distan …

Quoting the formula without deriving it from the two successive single-surface refractions, or missing the specific n=1.5-and-sy …