Given the series:

b, -3, 4b , -12 .

Let r be the common difference.

Then we know that:

-3 = b*r................(1)

4b = b*r^2.............(2)

-12 = b*r^3.................(3)

First we will rewrite (3).

-12 = br*r^2

But, from (1), we know that b*r = -3

==> -12 = -3*r^2

==> r^2 = -12/-3 = 4

==> r1 = 2 ==> br= -3 ==> 2b=-3 ==> **b1= -3/2**

==> r2= -2 ==> br = -3 ==> -2b = -3 ==> **b2 = 3/2**

**==> -3/2, -3, -6 , -12 are terms of a G.P where r = 2**

**==> 3/2 , -3, 6, -12 are terms of a G.P where r= -2**

We'll apply the theorem of the geometric mean and we'll get:

4b = sqrt (-3)*(-12)

4b = sqrt 36

4b = 6

b = 6/4

**b = 3/2**

**or**

**b = -3/2**

**For b = 3/2, we'll get the common ratio:**

r = -3*2/3

**r = -2**

**For b = -3/2, we'll get the common ratio:**

r = -3*2/-3

**r = 2**

What is b in the following series: b, -3 , 4b, -12 if it is a G.P.

Since the given series with 4 terms are in geometric progression (G.P), we get the successive terms a1,a2, ... etc as below:

a1 = b..................(1).

a2 = b*r = -3.......(2) .where r is the common ratio.

a3 = br^2 = 4b....(3).

a4 = br^3 = -12..(4).

(4)/(2): 2^2 = -12/-3 = 4. So r = sqrt4 = 2, orr= -sqrt4 = -2.

If r = 2, b = -3/3 = -1.5.

If r= -2, then b= -3/-2 = 1.5.

Therefore b = -1.5 if = 2. b = 1.5, if r = -2.