Suppose is a geometric sequence with a common ratio of where . If is an arithmetic sequence, then the sum of all possible values of can be expressed as . Find .
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Since x , y , and z are in a geometric sequence, we can express y as x r and z as x r 2 . Therefore, since 3 x , 2 6 y , and 1 7 z are in an arithmetic progression, we can construct the equation 1 7 z − 2 6 y = 2 6 y − 3 x . We can plug in the values of y and z in the equation to get 1 7 x r 2 − 2 6 x r = 2 6 x r − 3 x . We can eliminate the x terms to get 1 7 r 2 − 2 6 r = 2 6 r − 3 . From this, we can rearrange terms and get the quadratic 1 7 r 2 − 5 2 r + 3 = 0 . By Vieta's, the sum of the roots of a quadratic is equivalent to − a b , so the sum of all possible values of r is 1 7 5 2 . Our answer is 5 2 + 2 ( 1 7 ) = 8 6 .