Question 192687
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If the arithmetic progression has a common difference of <b><i>d</i></b>, and the first of the three numbers is <b><i>x</i></b>, then the second number must be <b><i>x + d</i></b> and the third, <b><i>x + </i>2</b><b><i>d</i></b>.


So if the sum is 21, then:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x + (x + d) + (x + 2d) = 21]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3x + 3d = 21]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x + d = 7]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x = 7 - d] is the first number,


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x + d = 7 - d + d = 7] is the second number, and 


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x + 2d = 7 - d + 2d = 7 + d] is the third number.


Now, if the product is 180, then:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (7 - d)(7)(7 + d) = 180]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 49 - d^2 = \frac{180}{7}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d^2 = \frac{163}{7}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d = \sqrt{\frac{163}{7}}]


Rationalizing the denominator:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d = \frac{\sqrt{1141}}{7}  ]


and the three numbers are:


*[tex \LARGE 7 - \frac{\sqrt{1141}}{7}],


*[tex \LARGE 7], and 


*[tex \LARGE 7 + \frac{\sqrt{1141}}{7}].




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