Find x if the numbers x, 6, x-5 are the terms of a geometric series.

The terms x,6,x-5 are the consecutive terms of a geometric
progression (series) if and only if the middle term is the geometric mean of the
neighbor terms:

6 =
sqrt[x*(x-5)]

We'll raise to square both
sides:

36 = x(x-5)

We'll use
the symmetric property and we'll remove the brackets:

x^2 -
5x = 36

We'll subtract 36:

x^2
- 5x - 36 = 0

We'll apply the quadratic
formula:

x1 = [5 + sqrt(25 +
144)]/2

x1 =
(5+13)/2

x1 =
9

x2 =
(5-13)/2

x2 =
-4

We'll check the
solution:

For x =
-4:

-4 , 6 , -9

6/-4 =
-3/2

-9/6 = -3/2

So, the
common ratio of the g.p. whose terms are -4 , 6 , -9, is r =
-3/2.

For x = 9

9 , 6 ,
4

6/9 = 2/3

4/6 =
2/3

So, the common ratio of the g.p. whose terms are 9 , 6
, 4 is r = 2/3.