Asked by Teagan
Find all values of x such that x-6, x+6, and 10 x -6 form a geometric sequence. Give your answers in increasing order
Answers
Answered by
Reiny
In a GS
(x+6)/(x-6) = (10x-6)/(x+6)
cross-multiply
(x+6)^2 = (x-6)(10x-6)
x^2 + 12x + 36 = 10x^2 -66x + 36
9x^2 - 78x = 0
x(9x-78) = 0
x = 0 or x = 78/9 = 26/3
if x = 0, then our terms would be -6, 6, and -6
which technically is a GS, with r = -1, just flipping between -6 and 6 , not very interesting.
if x = 26/3
terms are:
26/3 - 6, 26/3 + 6, and 260/3 - 6
or
8/3 , 44/3, and 242/3
(x+6)/(x-6) = (10x-6)/(x+6)
cross-multiply
(x+6)^2 = (x-6)(10x-6)
x^2 + 12x + 36 = 10x^2 -66x + 36
9x^2 - 78x = 0
x(9x-78) = 0
x = 0 or x = 78/9 = 26/3
if x = 0, then our terms would be -6, 6, and -6
which technically is a GS, with r = -1, just flipping between -6 and 6 , not very interesting.
if x = 26/3
terms are:
26/3 - 6, 26/3 + 6, and 260/3 - 6
or
8/3 , 44/3, and 242/3
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