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.