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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!

Determine the correct formula for the following geometric sequence: 18, 6, 2, 2/3, ...

Learn how to determine the formula for a geometric sequence where the first term is 18 and the common ratio is 1/3. Step-by-step explanation included for solving geometric sequences. Suitable for high school students.

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