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To identify the correct recursive equation for the table, let's analyze the pattern in the \( y \)-values.

Here’s the table dataTo identify the correct recursive equation for the table, let's analyze the pattern in the \( y \)-values.

Here’s the table 

\[
\begin{array}{c|c}
x & y \\
\hline
1 & 15 \\
2 & 18 \\
3 & 21 \\
\end{array}
\]

### Step 1: Identify the pattern
The values of \( y \) increase by a constant amount as \( x \) increases by 1. The differences are:
- From \( y_1 = 15 \) to \( y_2 = 18 \), the increase is \( 18 - 15 = 3 \).
- From \( y_2 = 18 \) to \( y_3 = 21 \), the increase is \( 21 - 18 = 3 \).

So, the pattern shows that each \( y \)-value increases by 3.

### Step 2: Recursive Formula
The recursive formula has the following structure:
- The first term, \( f(1) \), is given by the first value in the table, which is \( f(1) = 15 \).
- For \( n \geq 2 \), the function increases by 3, meaning \( f(n) = f(n-1) + 3 \).

### Step 3: Correct Answer
Now, let’s match this with the given choices:

1. \( f(n) = 15 + 3(n - 1) \) — This is not a recursive equation, it's a closed form.
2. \( f(1) = 15; f(n) = f(n-1) + 3 \) — This is a correct recursive formula.
3. \( f(1) = 15; f(n) = f(n-1) - 3 \) — Incorrect because it suggests decreasing by 3.
4. \( f(0) = 15; f(n) = f(n-1) + 3 \) — Incorrect starting point as it should start at \( f(1) \).

Thus, the correct recursive equation is:

\[
\boxed{f(1) = 15; f(n) = f(n-1) + 3}
\]

Would you like more details or have any questions?

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Here are 5 related questions for deeper understanding:
1. How would you write a closed-form equation for the same table?
2. What is the general form of a recursive equation for an arithmetic sequence?
3. How do you determine the initial condition for a recursive equation?
4. How would the recursive equation change if the difference between the values were -3 instead of +3?
5. Can you solve for \( f(4) \) and \( f(5) \) using the recursive equation?

**Tip:** Recursive equations are useful for defining sequences where each term depends on the previous one, often requiring an initial condition.

Identify the correct RECURSIVE equation for this table:

x | y
--|--
1 | 15
2 | 18
3 | 21

Options:
1. f(n) = 15 + 3(n - 1)
2. f(1) = 15; f(n) = f(n - 1) + 3
3. f(1) = 15; f(n) = f(n - 1) - 3
4. f(0) = 15; f(n) = f(n - 1) + 3

This problem asks to identify the correct recursive equation for the sequence where y-values increase by 3 as x increases. Understand recursive formulas and arithmetic sequence concepts through this example.

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