The problem asks to find the missing term in the given geometric sequence:

$6, \, \boxed{?}, \, \frac{3}{2}$

Steps to Solve:

1. Definition of a geometric sequence: In a geometric sequence, the ratio between consecutive terms remains constant. Let the common ratio be $r$.

2. Determine the ratio using known terms: The sequence is: $6, \, ?, \, \frac{3}{2}$ The second term (missing) is $6 \cdot r$, and the third term is $6 \cdot r^2 = \frac{3}{2}$.

3. Find $r$: From the equation $6 \cdot r^2 = \frac{3}{2}$: $r^2 = \frac{\frac{3}{2}}{6} = \frac{3}{12} = \frac{1}{4}$ $r = \pm \frac{1}{2}$

4. Find the missing term:

   • For $r = \frac{1}{2}$: $\text{Missing term} = 6 \cdot r = 6 \cdot \frac{1}{2} = 3$
   • For $r = -\frac{1}{2}$: $\text{Missing term} = 6 \cdot r = 6 \cdot \left(-\frac{1}{2}\right) = -3$

Final Answer:

The missing term is either $3$ (if $r = \frac{1}{2}$) or $-3$ (if $r = -\frac{1}{2}$).

Do you have any questions or want detailed steps for verification?

Related Questions:

1. What is the general formula for a geometric sequence?
2. How do we differentiate between arithmetic and geometric sequences?
3. What happens if the sequence alternates between positive and negative terms?
4. Can a geometric sequence have a common ratio of 0? Why or why not?
5. How does the choice of positive or negative $r$ affect the overall sequence?

Tip: Always check the given terms to decide whether the sequence alternates or stays consistent in sign!