What value of x makes the three terms x, x/(x + 1) , 3x/(x + 1)(x + 2) those of a geometric sequence?

If a, b and c are three consecutive terms of a gemetric  series,
then

 ac = b^2.

Similarly if  x, x/x+1
and 3x/(x+1)(x+2)  are the three consecutive terms of a geometric series,
then

x* 3x/(x+1)(x+2) =
{x/(x+1)}^2.

Now we multiply both sides by the LCM ,(x+1)^2*(x+2) of
denominators.

3x^2(x+1) =
x^2*(x+2)

Divide by x^2 both
sides:

3(x+1) = x+2.

3x+3 =
x+2.

3x-x = 2-1 = -1.

2x =
-1.

x = -1/2.

Therefore x = -1/2 makes
the the three terms in GP  :

First term x = -1/2 ,
 

2nd term x/(x+1) = -0.5/(-0.5+1) = -1
and

3rd term 3x/(x+1)(x+2) = -1.5/(0,5)(1.5) =  -2  which is in
geometrac ratio with common ratio  1/(-1/2) = -2/(-1) = 2.