Let’s solve the problem step by step:

Recursive Formula

The problem states that the number of cars sold weekly grows linearly. Based on the information:

• $P_0 = 6$ (Initial value, cars sold at the beginning)
• $P_1 = 8$ (Cars sold in the first week)

From $P_0$ and $P_1$, the growth rate (difference) is: $P_1 - P_0 = 8 - 6 = 2$ This means the dealership sells 2 more cars each week than the previous week.

The recursive formula is: $P_N = P_{N-1} + 2$

Explicit Formula

The explicit formula uses the initial value and growth rate: $P_N = P_0 + N \cdot \text{(growth rate)}$ Substitute $P_0 = 6$ and growth rate $2$: $P_N = 6 + 2N$

Number of Cars in Fourth Week

Using the explicit formula $P_N = 6 + 2N$, substitute $N = 4$ (fourth week): $P_4 = 6 + 2 \cdot 4 = 6 + 8 = 14$ Thus, 14 cars will be sold in the fourth week.

Final Answers

1. Recursive Formula: $P_N = P_{N-1} + 2$
2. Explicit Formula: $P_N = 6 + 2N$
3. Cars in Fourth Week: $14$

Would you like additional details or explanations? Here are some related questions to expand your understanding:

1. What is the difference between recursive and explicit formulas?
2. How can you calculate the growth rate if only two weeks' data are given?
3. How would the formula change if the growth rate were negative?
4. Can you graph the linear growth model using these formulas?
5. What would the total number of cars sold over four weeks be?

Tip: Always verify formulas by testing known values to ensure correctness!