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

$a_1=\frac{12}{r}$

We replace this in the second equation

$108=\frac{12}{r}\cdot r^3$

Now, we solve for r

$\begin{gathered} 108=12r^2 \\ \frac{12r^2}{12}=\frac{108}{12} \\ r^2=9 \\ r=\sqrt[]{9} \\ r=3 \end{gathered}$

Therefore, the common ratio is 3.