Based on the uploaded image, here is the breakdown of the problem and the steps to solve it:

Problem:

The number of cars sold weekly by a dealership grows according to a linear growth model. The details are:

• First week: $P_0 = 6$ cars sold.
• Second week: $P_1 = 14$ cars sold. You are tasked to:

1. Write a recursive formula for $P_N$, the number of cars sold in the $N+1$-th week.
2. Write an explicit formula for $P_N$.
3. Calculate how many cars will be sold in the fourth week.

1. Recursive Formula:

To find the recursive formula, we need the difference in sales between consecutive weeks: $P_1 - P_0 = 14 - 6 = 8.$ This shows the sales increase by 8 cars each week. The recursive formula is: $P_N = P_{N-1} + 8.$

2. Explicit Formula:

The explicit formula for linear growth is: $P_N = P_0 + N \cdot d,$ where:

• $P_0 = 6$ (initial sales),
• $d = 8$ (weekly increase).

Thus, the explicit formula is: $P_N = 6 + 8N.$

3. Cars Sold in the Fourth Week:

To find sales in the fourth week ($P_3$), use the explicit formula: $P_3 = 6 + 8 \cdot 3 = 6 + 24 = 30.$

Final Answers:

1. Recursive Formula: $P_N = P_{N-1} + 8$.
2. Explicit Formula: $P_N = 6 + 8N$.
3. Sales in the Fourth Week: $30$ cars.

Let me know if you need further clarification or details!

Follow-Up Questions:

1. How would the formulas change if the growth rate wasn't constant?
2. Can you model the growth using exponential growth instead?
3. What would happen if sales decreased each week instead of increasing?
4. How do recursive and explicit formulas relate in general for growth models?
5. How would you plot this growth over time?

Tip:

Always double-check the difference between consecutive values to confirm the growth type (linear or exponential).