Question

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?

Answer 1

For a geometric sequence, the ratio between the third and second term is the same as the ratio between the fourth and third term.

Thus, we can say:
`12x = r` and
`r \cdot x = (4)/(3)`

`12x = (4)/((3)/(x))}`

`12x = (4)/(3x)`

`36x^(2) = 4`

`9x^(2) = 1`

`x^(2) = (1)/(9)`

`x^(2) = \pm (1)/(3)`

Hence, the common ratio for this sequence is:
`r = \pm (1)/(3)`

Answer 2

1/3

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.