The sequence provided is $18, 6, 2, \frac{2}{3}, \dots$. This is a geometric sequence where each term is obtained by multiplying the previous term by a common ratio.

Step 1: Identify the common ratio.

To find the common ratio $r$, divide the second term by the first term: $r = \frac{6}{18} = \frac{1}{3}$ Thus, the common ratio is $\frac{1}{3}$.

Step 2: Use the geometric sequence formula.

The general formula for the $n$-th term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ where:

• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the term number.

For this sequence:

• $a_1 = 18$,
• $r = \frac{1}{3}$.

Thus, the formula becomes: $a_n = 18 \cdot \left( \frac{1}{3} \right)^{n-1}$

Step 3: Compare with the provided options.

The correct answer matches the formula $a_n = 18 \cdot \left( \frac{1}{3} \right)^{n-1}$, which is the first option.

Final Answer:

The correct formula is: $\boxed{a_n = 18 \left( \frac{1}{3} \right)^{n-1}}$

Do you need further details or clarifications? Here are some related questions to deepen your understanding:

1. How is the common ratio found in a geometric sequence?
2. How can you identify if a sequence is geometric?
3. What happens if the common ratio is negative in a geometric sequence?
4. Can a geometric sequence have a non-integer common ratio?
5. How does the value of the first term affect the geometric sequence?

Tip: For geometric sequences, knowing the common ratio and the first term allows you to predict any term in the sequence!