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The table in the image presents a geometric sequence. To solve this, we'll first find the missing values in the sequence and then write the explicit and recursive equations.

### Step 1: Identify the common ratio
In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio (\(r\)).

The terms given are:
- \( f(1) = 2 \)
- \( f(2) = 6 \)
- \( f(3) = 18 \)
- \( f(4) = 54 \)
- \( f(5) = 162 \)

Now, divide consecutive terms to find the common ratio:
\[
r = \frac{6}{2} = 3 \quad \text{and} \quad r = \frac{18}{6} = 3
\]
Thus, the common ratio is \( r = 3 \).

### Step 2: Fill in missing terms
Given \( r = 3 \), we can check all terms:
- \( f(2) = f(1) \times 3 = 2 \times 3 = 6 \)
- \( f(3) = f(2) \times 3 = 6 \times 3 = 18 \)
- \( f(4) = f(3) \times 3 = 18 \times 3 = 54 \)
- \( f(5) = f(4) \times 3 = 54 \times 3 = 162 \)

Thus, all the terms are correct, and no additional missing terms need to be filled.

### Step 3: Write the explicit and recursive equations
- **Explicit Equation**: For a geometric sequence, the explicit formula is:
  \[
  f(n) = f(1) \times r^{n-1}
  \]
  Substituting \( f(1) = 2 \) and \( r = 3 \):
  \[
  f(n) = 2 \times 3^{n-1}
  \]

- **Recursive Equation**: The recursive formula for a geometric sequence is:
  \[
  f(n) = f(n-1) \times r \quad \text{with} \quad f(1) = 2
  \]
  Substituting \( r = 3 \):
  \[
  f(n) = f(n-1) \times 3 \quad \text{with} \quad f(1) = 2
  \]

### Final Answer:
- **Explicit Equation**: \( f(n) = 2 \times 3^{n-1} \)
- **Recursive Equation**: \( f(n) = f(n-1) \times 3 \) with \( f(1) = 2 \)

Do you need further details or clarification?

Here are 5 related questions:
1. How do you find the common ratio in a geometric sequence?
2. What are the differences between explicit and recursive equations?
3. Can a geometric sequence have a negative common ratio?
4. How do you determine if a sequence is geometric or arithmetic?
5. What happens if the common ratio is a fraction?

**Tip**: When writing recursive formulas, always define the first term separately.

Find the missing terms of the geometric sequence represented in the table, and then write the explicit and recursive equations for the sequence.

Learn how to find the missing terms of a geometric sequence and write explicit and recursive equations. This problem includes a step-by-step solution covering common ratios, and formulas for geometric sequences. Suitable for Grades 8-10.

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