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With lenses the formula for the focal point is again the reciprocal of the sum of the reciprocals of the distances to the image and to the object.

Is there some underlying physics principle behind this formula that allows it to show up in different places? I got that in general the total is inversely related to the number of terms, but is there a stronger connection here?

tl;dr Why does this formula show up in different places, 1/x + 1/x = 1/y ?

thank you for any insight.