5. Andrea and Kevin were working on the following arithmetre sequence: $$ 4,7,10,13,16 \text {. . } $$ a. Andrea wrote the following explicit equation for the sequence: $$ r(n)=3 n+2 $$ Is Andrea correct? Explain why or why not. If not, write a correct explicit equation for the sequence. K. Kevin wrote the following recursive equation for the sequence: t( $$ n+1 $$ ) = t(n) +3 Is Kevin correct? Explain why or why not. If not, write a correct recursive cquation for the sequence.

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To solve the problem, we need to analyze the arithmetic sequence given and the equations proposed by Andrea and Kevin.

### Step 1: Analyze the Arithmetic Sequence

The sequence provided is:
$$ 4, 7, 10, 13, 16, \ldots $$

**Common Difference:**
To find the common difference $$ d $$, we subtract the first term from the second term:
$$ d = 7 - 4 = 3 $$

This common difference is consistent throughout the sequence:
- $$ 10 - 7 = 3 $$
- $$ 13 - 10 = 3 $$
- $$ 16 - 13 = 3 $$

### Step 2: Evaluate Andrea's Explicit Equation

Andrea's explicit equation is:
$$ r(n) = 3n + 2 $$

**Check the Equation:**
To see if this equation correctly represents the sequence, we can calculate the first few terms using $$ n = 1, 2, 3, \ldots $$:

- For $$ n = 1 $$:
  $$ r(1) = 3(1) + 2 = 5 $$
- For $$ n = 2 $$:
  $$ r(2) = 3(2) + 2 = 8 $$
- For $$ n = 3 $$:
  $$ r(3) = 3(3) + 2 = 11 $$

The values $$ 5, 8, 11 $$ do not match the sequence $$ 4, 7, 10, 13, 16 $$.

**Conclusion for Andrea:**
Andrea's equation is incorrect.

### Step 3: Write the Correct Explicit Equation

The general form of an explicit equation for an arithmetic sequence is:
$$ r(n) = a + (n-1)d $$
where $$ a $$ is the first term and $$ d $$ is the common difference.

Here, $$ a = 4 $$ and $$ d = 3 $$:
$$ r(n) = 4 + (n-1) \cdot 3 $$
$$ r(n) = 4 + 3n - 3 $$
$$ r(n) = 3n + 1 $$

### Step 4: Evaluate Kevin's Recursive Equation

Kevin's recursive equation is:
$$ t(n+1) = t(n) + 3 $$

**Check the Equation:**
To see if this equation correctly represents the sequence, we need to define the first term:
Let $$ t(1) = 4 $$.

Now we can calculate the next terms:
- For $$ n = 1 $$:
  $$ t(2) = t(1) + 3 = 4 + 3 = 7 $$
- For $$ n = 2 $$:
  $$ t(3) = t(2) + 3 = 7 + 3 = 10 $$
- For $$ n = 3 $$:
  $$ t(4) = t(3) + 3 = 10 + 3 = 13 $$
- For $$ n = 4 $$:
  $$ t(5) = t(4) + 3 = 13 + 3 = 16 $$

The values $$ 4, 7, 10, 13, 16 $$ match the sequence.

**Conclusion for Kevin:**
Kevin's equation is correct.

### Final Answers

- **Andrea's Equation:** Incorrect. The correct explicit equation is:
  $$ r(n) = 3n + 1 $$

- **Kevin's Equation:** Correct. The recursive equation is:
  $$ t(n+1) = t(n) + 3 $$ with $$ t(1) = 4 $$.
Andrea's explicit equation $$ r(n) = 3n + 2 $$ is incorrect. The correct explicit equation is $$ r(n) = 3n + 1 $$. Kevin's recursive equation $$ t(n+1) = t(n) + 3 $$ is correct.

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