The second term in a geometric sequence is 12. the fourth term in the same sequence is 4/3. what is the common ratio in this sequence?

1/3

Step-by-step explanation:

Given:-

For the geometric sequence:

- 2nd Term = 12

- 4th Term = 4 / 3

Find:-

What is the common ratio in this sequence?

Solution:-

- A geometric sequence is categorized by two parameters which are:

                                  a : First Term

                                  r : Common ratio.

- The general (nth) term in a geometric sequence is given by the following relation.

                           Term = a * ( r ) ^ ( n - 1 )

Where,                n = Term number

- We will develop two equations for the given two terms as follows:

                           12 = a * ( r ) ^ ( 2 - 1 ) = a * ( r )

                           4/3 = a * ( r ) ^ ( 4 - 1 ) = a * ( r )^3

- Now divide the two equations and solve for common ratio (r):

                           12*3 / 4 = 1 / r^2

                           9 =  1 / r^2

                           r = √(1 /9) = 1 / 3

- The common ratio (r) is equal to 1 / 3.

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!