Question 221922
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Let *[tex \Large x] represent the first integer.  Then *[tex \Large x + 2] is the next consecutive even integer.  The reciprocal of the first integer is then *[tex \Large \frac{1}{x}] and the reciprocal of the second is *[tex \Large \frac{1}{x+2}]


The sum of the reciprocals is *[tex \Large \frac{7}{24}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{x}\ +\  \frac{1}{x+2}\ = \frac{7}{24}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{2x+2}{x^2+2x}\ = \frac{7}{24}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 7x^2\ -\ 34x\ -\ 48\ =\ 0]


The quadratic factors, but one of the roots is not an integer.  Exclude the non-integer root.  The integer root is the smaller of your two integers.  Add 2 to get the other one.



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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