I think the formula in comparing image size of 2 lenses involves squaring each lens focal length then dividing. For example in comparing image size of a 500 & 600mm lens > (600x600)/(500x500)=360000/250000=1.44. So a 600mm lens will produce an image 1.44 times larger than a 500mm lens at the same distance.
I think this is correct & is essentially what you are saying (with respect to a 500 & a 600 lens) but perhaps someone can confirm or refute this.
NOTE if comparing a 200mm & a 300mm lens > 9/4= 2.25.