The problem asks for the possible values of the common ratio in a geometric sequence. Here's a structured way to solve it:

Given:

• The 7th term of the geometric sequence ($a_7$) is 13.
• The 9th term of the geometric sequence ($a_9$) is 52.

General Formula for a Geometric Sequence:

$a_n = a_1 \cdot r^{n-1}$ where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $r$ is the common ratio.

We know the following:

• $a_7 = a_1 \cdot r^6 = 13$
• $a_9 = a_1 \cdot r^8 = 52$

Step 1: Divide the two equations

Dividing the two equations will eliminate $a_1$, leaving us with an equation involving only $r$.

$\frac{a_9}{a_7} = \frac{a_1 \cdot r^8}{a_1 \cdot r^6} = r^2$ Substitute the known values: $\frac{52}{13} = r^2$ $r^2 = 4$ Thus, the possible values for $r$ are: $r = 2 \quad \text{or} \quad r = -2$

Conclusion:

The common ratio $r$ could be $2$ or $-2$.

Would you like more details on any step, or have additional questions?

Here are five related questions to further your understanding:

1. How does the common ratio affect the terms of a geometric sequence?
2. What happens to the terms if the common ratio is negative?
3. Can a geometric sequence have a common ratio of 0?
4. What is the difference between an arithmetic and geometric sequence?
5. How would you find the first term $a_1$ given more information?

Tip: When dividing equations to solve for the common ratio, make sure that both terms have been expressed in terms of the same powers of $r$ to simplify calculations.