# Calculate the real number x>=4, when 2, x-4, x are the consecutive terms of a geometric progression.

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2, x-4, x are terms of a geometric progression.

Then,

x-4 = 2r....(1)

and x= (x-4)*r...(2)

from (2) :

x= xr -4r

But from (1) we know that x-4=2r

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

==> x= x(x-4)/2 -4(x-4)/2

==> x= (1/2)x^2 -2x -2x +8

Multiply by 2:

==> 2x = x^2 -8x +16

==> x^2 -10x +16=0

Factorize:

(x-8)(x-2)=0

x1= 8

x2=2

But since x=>4 then we will ignore x2=2 as a solution

Then x=8

Then r= (x-4)/2= (8-4)/2 = 2

Let us check the answer:

The progression 2, x-4, x

==> 2, 4, 8 Which is a geometric progression with r=2

If 2, x-4, x are consecutive terms of the geometric progression, that means that:

x-4 = 2q, where q is the ratio

We'll divide by 2 and we'll get:

q = (x-4)/2 (1)

x = q*(x-4)

We'll divide by (x-4)

q = x/(x-4) (2)

From (1) and (2) it results:

(x-4)/2 = x/(x-4)

We'll cross multiplying:

(x-4)^2 = 2x

We'll expand the square:

x^2 - 8x + 16 = 2x

We'll move all terms to one side:

x^2 - 8x + 16 - 2x = 0

x^2 - 10x + 16 = 0

We'll apply the quadratic formula:

x1 = [10+sqrt(100-64)]/2

x1 = (10+6)/2

x1 = 16/2

x1 = 8

x2 = (10-6)/2

x2 = 2

Because, from the enunciation, x has to be more than 4, or at least 4, the second solution is not convenient. So, the only solution of the equation is x = 8.

In a geometric progression the ratio of any consecutive terms remains the same.

In the given geometric progression three consecutive terms are:

2, (x - 4), and x.

Therefore:

(x - 4)/2 = x/(x - 4)

Cross multiplying the terms in the above equation we get:

(x - 4)(x - 4) = 2*x

x^2 - 8x + 16 = 2x

x^2 - 8x - 2x + 16 = 0

x(x - 8) - 2(x - 8) = 0

(x - 8)(x - 2) = 0

Therefore x = 8 or x = 2

As the required value of x is >=4, the correct value of x is:

x = 8

2,x-4 andx are inGP. To find the solution such that x > 4.

Solution:

Since 2,x-4 and x are in GP, the the successive terms bear the same common ratio . So (x-4)/2 = x/(x-4). Or by cross multiplication, we get:

(x-4)^2 = 2x.

x^2-8x+16 = 2x. Or

x^2-8x-16-2x = 0. )r

x^2-10x +16 = 0. Or

(x-8)(x-2) =0.

So x-8 = 0 Or x = 8 is the solution as given that x>=4.