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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?

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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.

The seventh term of a geometric sequence is 13. The ninth term is 52. What values could the common ratio take?

Learn how to solve a geometric sequence problem where the seventh term is 13 and the ninth term is 52. This problem uses the formula for geometric sequences and explores the relationship between the terms to find the common ratio. Suitable for students in Grades 9-11.

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