Let's solve the problem step by step.

Recursive Formula for $P_N$

From the given information:

• The initial number of cars sold in week 0 is $P_0 = 6$.
• In week 1, the dealership sold $P_1 = 8$.

The growth in the number of cars sold weekly is constant: $P_1 - P_0 = 8 - 6 = 2$ This means the number of cars sold increases by 2 cars every week.

Thus, the recursive formula is: $P_N = P_{N-1} + 2$

Explicit Formula for $P_N$

The explicit formula of a linear sequence is given by: $P_N = P_0 + N \cdot d$ where:

• $P_0 = 6$ (initial value),
• $d = 2$ (common difference).

Substitute these values: $P_N = 6 + 2N$

Number of Cars Sold in the Fourth Week ($P_4$)

Using the explicit formula $P_N = 6 + 2N$, substitute $N = 4$: $P_4 = 6 + 2(4) = 6 + 8 = 14$

Final Answers

1. Recursive formula: $P_N = P_{N-1} + 2$
2. Explicit formula: $P_N = 6 + 2N$
3. Cars sold in the fourth week: 14 cars

If you have any questions or need more clarification, let me know!

Here are 5 relative questions for practice:

1. How would the recursive formula change if the growth was $3$ cars per week instead of $2$?
2. Derive the explicit formula if $P_0 = 10$ and the dealership sells $4$ more cars weekly.
3. What is the total number of cars sold after 5 weeks using the explicit formula?
4. How would you determine the week when the dealership sells $20$ cars?
5. Explain the difference between recursive and explicit formulas in this context.

Tip: Always check the difference between terms to ensure you correctly identify the growth rate for sequences!