Find the value of x if the geometric mean of 2x and 19x-2 is 7x-2

since the geometric mean is 7x-2 then we can say that the sequence with 3 terms would be:

2x, 7x-2, 19x-2

for it to be considered as a geometric sequence, it should have a common ratio. In which case we define the equation to find x as

(19x-2)/(7x-2) = (7x-2)/2x

cross multiply

(19x-2)(2x) = (7x-2)(7x-2)

distribute

38x^2-4x = 49x^2-14x-14x+4

simplify

38x^2-4x = 49x^2-14x-14x+4

49x^2-14x-14x+4-38x^2+4x=0

11x^2-24x+4 = 0

factor

(11x-2)(x-2)=0

11x-2=0

11x=2

x=2/11

and

x-2=0

x=2

therefore the values of 'x' would be 2/11 and 2

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