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The given sequence is:

\[
-\frac{2}{3}, -4, -24, -144, \dots
\]

We need to identify the formula that describes this sequence.

### Step 1: Identify the pattern in the sequence
Let's observe the ratio between consecutive terms:
- The second term is \(-4\), and the first term is \(-\frac{2}{3}\).
\[
\frac{-4}{-\frac{2}{3}} = 6
\]
- The third term is \(-24\), and the second term is \(-4\).
\[
\frac{-24}{-4} = 6
\]
- The fourth term is \(-144\), and the third term is \(-24\).
\[
\frac{-144}{-24} = 6
\]

The common ratio is \(6\), so this is a geometric sequence with a common ratio \(r = 6\).

### Step 2: General form of a geometric sequence
The general form of a geometric sequence is:
\[
f(x) = a \cdot r^{x-1}
\]
Where:
- \(a\) is the first term,
- \(r\) is the common ratio, and
- \(x\) is the term number.

Here:
- \(a = -\frac{2}{3}\),
- \(r = 6\).

Thus, the formula for the sequence is:
\[
f(x) = -\frac{2}{3} \cdot 6^{x-1}
\]

### Step 3: Match with the given options
The correct option that matches the formula is:

\[
f(x) = -\frac{2}{3} \cdot (6)^{x-1}
\]

This corresponds to the third option.

---

Would you like more details or have any further questions?

### Related Questions:
1. What is the difference between arithmetic and geometric sequences?
2. How do you find the common ratio in a geometric sequence?
3. Can geometric sequences have negative common ratios?
4. What happens to a geometric sequence if the common ratio is a fraction?
5. How would you find the sum of the first \(n\) terms in a geometric sequence?

**Tip**: In geometric sequences, multiplying by the common ratio repeatedly generates the next terms.

Which formula can be used to describe the sequence: -2/3, -4, -24, -144,...?

Learn how to find the formula for the geometric sequence -2/3, -4, -24, -144. This sequence has a common ratio of 6, and the formula follows the geometric sequence pattern. Suitable for high school students in Grades 9-12.

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