# Geometric SeriesFind x if the numbers x, 6, x-5 are the terms of a geometric series.

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The numbers x, 6 and x - 5 are terms of a geometric series. The consecutive terms of a geometric series have a common ratio.

(x - 5)/6 = 6/x

=> x(x - 5) = 36

=> x^2 - 5x = 36

=> x^2 - 5x - 36 = 0

=> x^2 - 9x + 4x - 36 = 0

=> x(x - 9) + 4(x - 9) = 0

=> (x + 4)(x - 9) = 0

x can be -4 or 9

**The value of x is (-4 , 9)**

We'll use the mean theorem of a geometric sequence:

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

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