Generating Converging Whole Numbers for (2x+3y)/(x+y) = e

[Loren Booda]

How can one generate the sequence of whole numbers x and y which converge upon the equality

(2x+3y)/(x+y) = e

?

 

[AKG]

(2x+3y)/(x+y) = e gives x/y = (e-3)/(2-e), so it suffices to find a sequence of rationals which converges to (e-3)/(2-e). If the decimal expansion of (e-3)/(2-e) is given, then we can just set y = 1, 10, 100, 1000, 10000, ... and make the obvious choice for x.

 

[christianjb]

(2x+3y)/(x+y) = e gives x/y = (e-3)/(2-e), so it suffices to find a sequence of rationals which converges to (e-3)/(2-e). If the decimal expansion of (e-3)/(2-e) is given, then we can just set y = 1, 10, 100, 1000, 10000, ... and make the obvious choice for x.

Same as above- except use continued fractions for x/y=(e-3)/(2-e)