What is a formula for the nth term of the given sequence?
192, comma, minus, 144, comma, 108, point, point, .


The given sequence is:  
\[ 192, -144, 108, \ldots \]

To find the formula for the \(n\)-th term of the sequence, we need to identify the pattern or rule governing the sequence. Let's observe the changes between the terms:

1. **Identify the pattern:**  
   Each term appears to be multiplying by a specific ratio or number. Let's find this ratio by dividing consecutive terms:

\[
\frac{-144}{192} = -\frac{3}{4}, \quad \frac{108}{-144} = -\frac{3}{4}
\]

2. **Common ratio:**  
   The common ratio between the terms is \(-\frac{3}{4}\).  

3. **Formula for the nth term of a geometric sequence:**  
   The general formula for the \(n\)-th term \(a_n\) of a geometric sequence is:

\[
a_n = a_1 \cdot r^{n-1}
\]

where:  
- \(a_1\) is the first term (192 in this case)  
- \(r\) is the common ratio (-3/4 in this case)  

4. **Substitute values:**  

\[
a_n = 192 \cdot \left( -\frac{3}{4} \right)^{n-1}
\]

### Final Answer:
The formula for the \(n\)-th term of the sequence is:  
\[
\boxed{a_n = 192 \cdot \left( -\frac{3}{4} \right)^{n-1}}
\]

Would you like a deeper explanation, or do you have any other questions?

### Related Questions:
1. What is a geometric sequence, and how do you identify one?
2. How do you find the sum of the first \(n\) terms of a geometric sequence?
3. Can the formula be simplified further in any cases?
4. What happens to the terms as \(n\) becomes very large in this sequence?
5. How can you determine whether a sequence is convergent or divergent?

**Tip:** For geometric sequences, pay attention to the ratio between consecutive terms; it reveals the multiplier pattern.

What is a formula for the nth term of the given sequence? 192, comma, minus, 144, comma, 108, point, point, .

Discover how to find the nth term formula for the geometric sequence 192, -144, 108... using the common ratio and general formula for geometric sequences. This step-by-step solution is suitable for students in Grades 9-11, covering key concepts in geometric sequences.

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