# Find the value of x for which x+9, x-6, 4 are the firs three terms of a G.P. and calculate the fourth term of the progression.

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x+9, x-6, 4 are terms in a G.P

==> x-6 = (x+9)*r ==> r= (x+9)/(x-6)

==> 4= (x-6)*r ==> r= 4/(x-6)

==> (x-6)^2 = 4(x+9)

==> x^2 - 12x + 36 = 4x + 36

==> x^2 -16x = 0

==> x(x-16) = 0

==> x1= 0

==> x2= 16

For x1= 0

==> r= 4/(x-6) = 4/6 = 2/3

==> a1= x+9 = 9

==> a2= 6

==> a3= 9*4/9= 4

==> **a4= 9*8/27= 8/3**

For x2= 16

==> r= 4/(x-6) = 4/10 = 2/5

==> a1= 16+9 = 25

==> a2= 25*2/5= 10

a3= 25*4/25 = 4

**a4= 25*8/125 = 8/25**

==>

If x+9, x-6, 4 are the first 3 terms of a g.p., then:

(x-6)^2 = 4(x+9)

We'll expand the square:

x^2 - 12x + 36 = 4x + 36

We'll eliminate like terms:

x^2 - 12x - 4x + 36 - 36 = 0

x^2 - 16x = 0

We'll factorize:

x(x-16) = 0

We'll put each factor equal to zero:

x = 0

x-16 = 0

x = 16

For x = 0, the terms of the progression are: 9 , -6 , 4 , .....

We'll calculate the common ratio:

-6/9 = r

**-2/3 = r**

4/-6 = -2/3 = r

The fourth term is:

a4 = a*r^3

a4 = 9*(-2/3)^3

a4 = 9*(-8/27)

**a4 = -8/3**

For x = 16, we'll have the terms of the g.p.:

16+9, 16-6, 4 , ....

25 , 10 , 4

We'll calculate the common ratio:

r = 10/25

r = 2/5

r = 4/10

**r = 2/5**

a4 = a1*(r)^3

a4 = 25*(2/5)^3

a4 = 25*(8/25*5)

**a4 = 8/5**

Since x+9 ,x-6 and 4 are in GP, the product of 1st and last terms = square of the middle term.

(x+9)4 = (x-6)^2

4x+36 = x^2-12x+36

x^2-12x-4x = 0

x^2-16x = 0

x(x-16) = 0

x =0 or x= 16.

If x = 0, then thetrems are x+9 = 9, x-6 = -6 and 4 .

The common ratio is -6/9 = 4/-6 = -2/3. So the 4th term = 4*(-2/3) = -8/3

If x= 16, then the terms in order are:

x+9 =25, x-6 = 10 and the 3rd term is 4

The common ratio = 10/25 = 4/10 = 2/5. So the 4th term = 3rd term* common ratio = 4*2/5 = 8/5.