To find the common ratio $r$ of the geometric sequence, we can use the information given:

• The third term $a_3 = -108$
• The eighth term $a_8 = 26,244$

For a geometric sequence, the $n$-th term is given by the formula: $a_n = a \cdot r^{n-1}$ where $a$ is the first term, and $r$ is the common ratio.

Step 1: Set up equations

Using the third term: $a \cdot r^{2} = -108$ Using the eighth term: $a \cdot r^{7} = 26,244$

Step 2: Divide the equations to eliminate $a$

Dividing the second equation by the first: $\frac{a \cdot r^{7}}{a \cdot r^{2}} = \frac{26,244}{-108}$ This simplifies to: $r^{5} = \frac{26,244}{-108}$

Step 3: Calculate $r^{5}$

$r^{5} = -243$

Step 4: Find $r$ by taking the fifth root

The fifth root of $-243$ is $-3$.

Answer

The common ratio $r$ is $\mathbf{-3}$.

Let me know if you'd like more details or have any questions!

Here are some related questions to practice:

1. How would you find the first term of this sequence given the third term and common ratio?
2. What would the tenth term of this sequence be?
3. How do you determine if a sequence is geometric?
4. What is the formula to sum the first $n$ terms of a geometric sequence?
5. How would the solution change if the eighth term were negative?

Tip: In a geometric sequence, dividing two consecutive terms gives the common ratio.