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.