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.