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

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The discussion focuses on generating whole number sequences for the equation (2x+3y)/(x+y) = e, where e represents Euler's number. It establishes that the ratio x/y can be expressed as (e-3)/(2-e), allowing for the derivation of rational sequences that converge to this value. By setting y to powers of ten (1, 10, 100, etc.), participants suggest a straightforward method for determining corresponding x values. Additionally, the use of continued fractions is proposed as an alternative approach for achieving convergence.

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

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

?
 
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(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.
 
AKG said:
(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)
 

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