Question

A geometric sequence has a second term of 12 and a fourth term of 108. Find the possible values of the common ratio.

Answer 1

We know that

• The second term is 12.

,

• The fourth term is 108.

Geometric sequences are defined by

`a_n=a_1\cdot r^(n-1)`

Where,

`\begin{gathered} a_2=12 \\ a_4=108 \end{gathered}`

As you can observe, we don't have the first term of the sequence. So, we have to form a system of equations with the given information, that way we will be able to find the answer.

`\begin{gathered} 12=a_1\cdot r^(2-1)\rightarrow12=a_1\cdot r^{} \\ 108=a_1\cdot r^(4-1)\rightarrow108=a_1\cdot r^3 \end{gathered}`

We can solve the first equation for a1

`a_1=(12)/(r)`

We replace this in the second equation

`108=(12)/(r)\cdot r^3`

Now, we solve for r

`\begin{gathered} 108=12r^2 \\ (12r^2)/(12)=(108)/(12) \\ r^2=9 \\ r=\sqrt[]{9} \\ r=3 \end{gathered}`

Therefore, the common ratio is 3.