To solve this problem, we will analyze the given sequence and determine its general term.

Given Sequence:

$\frac{2}{3}, \frac{2}{9}, \frac{2}{27}, \frac{2}{81}, \frac{2}{243}, \dots$

Step 1: Observe the pattern of denominators

The denominators are: $3, 9, 27, 81, 243, \dots$ These are powers of 3: $3 = 3^1, \quad 9 = 3^2, \quad 27 = 3^3, \quad 81 = 3^4, \quad 243 = 3^5, \dots$ So, the denominator of the $n$-th term is $3^n$.

Step 2: Observe the pattern of numerators

The numerator in every term is $2$. Thus, the numerator is constant.

Step 3: General formula

The general term $a_n$ is a fraction: $a_n = \frac{\text{numerator}}{\text{denominator}} = \frac{2}{3^n}.$

Alternatively, this can be written as: $a_n = 2 \cdot \left(\frac{1}{3}\right)^n.$

Step 4: Adjust for the given first term

In the given sequence, $a_1 = \frac{2}{3}$, and this corresponds to $n=1$. Hence, we rewrite the formula as: $a_n = 2 \cdot \left(\frac{1}{3}\right)^{n-1}.$

Final Answer:

The general term is: $a_n = 2 \cdot \left(\frac{1}{3}\right)^{n-1}.$

This matches the third option in the provided choices.

Do you want further clarification or a breakdown of the steps? Let me know!

Here are five related questions to explore further:

1. How can you test if a formula is correct for a sequence?
2. How would the formula change if the numerator was $n^2$ instead of 2?
3. What is the sum of the first $k$ terms of this sequence?
4. How does the formula for a geometric sequence generalize?
5. What happens if the sequence starts with a different first term?

Tip: Always confirm the pattern by substituting the first few terms into your formula!