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Let's analyze the table you provided. We have the following values for \(x\) and \(y\):

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 3 \\
2 & \quad \\
3 & 108 \\
4 & \quad \\
\hline
\end{array}
\]

### Step-by-step reasoning for determining the explicit equation (part b):

The relationship between \(x\) and \(y\) seems to be exponential or multiplicative, as the value of \(y\) increases significantly between \(x = 1\) and \(x = 3\). We can try an exponential model of the form:

\[
y = a \cdot r^{x-1}
\]

where:
- \(a\) is the initial value (the value when \(x = 1\)),
- \(r\) is the common ratio between consecutive terms.

### Given:
- \(y(1) = 3\)
- \(y(3) = 108\)

Substitute \(x = 1\) into the equation:

\[
3 = a \cdot r^{1-1} = a
\]
Thus, \(a = 3\).

Now substitute \(x = 3\) and \(y = 108\):

\[
108 = 3 \cdot r^{3-1} = 3 \cdot r^2
\]

Solve for \(r\):

\[
r^2 = \frac{108}{3} = 36
\]
\[
r = 6
\]

### Therefore, the explicit equation is:
\[
y = 3 \cdot 6^{x-1}
\]

---

### Recursive equation (part c):

For a recursive relationship, we define \(y(x+1)\) in terms of \(y(x)\). Based on the explicit equation, we notice that each \(y(x+1)\) is obtained by multiplying the previous value of \(y(x)\) by 6. So, the recursive equation can be written as:

\[
y(x+1) = 6 \cdot y(x)
\]
with the initial condition \(y(1) = 3\).

---

Would you like more details or have any questions?

Here are 5 related questions to expand on this information:

1. What is the value of \(y\) when \(x = 2\) based on the explicit equation?
2. How can you verify that the recursive equation generates the same sequence as the explicit equation?
3. How would the explicit equation change if the common ratio \(r\) were different?
4. How can you use logarithms to solve for \(x\) if \(y\) is given in the explicit equation?
5. What is the difference between an arithmetic sequence and the geometric sequence used here?

**Tip:** When working with recursive equations, always remember to identify the initial condition.

Given the values in the table, find the explicit and recursive equations.

Learn how to find both explicit and recursive equations for an exponential sequence given in a table format. This problem involves calculating the common ratio and defining the recursive relation, suitable for students in Grades 9-10.

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