Problem Breakdown:

1. Linear Growth: The number of cars sold grows linearly, which means we can express the growth as an arithmetic sequence.
2. Week 1: 4 cars were sold.
3. Week 2: 11 cars were sold.

We are asked to find:

• The recursive formula for the number of cars sold in the $n$-th week.
• The explicit formula for the number of cars sold in the $n$-th week.
• The number of cars sold in the 4th week.

1. Finding the Recursive Formula:

For linear growth, the recursive formula follows the pattern:

$a_n = a_{n-1} + d$

Where:

• $a_n$ is the number of cars sold in the $n$-th week.
• $a_{n-1}$ is the number of cars sold in the previous week.
• $d$ is the common difference (the change in the number of cars sold each week).

We can find $d$ (the difference between consecutive weeks) using the values given: $d = 11 - 4 = 7$

Thus, the recursive formula is: $a_n = a_{n-1} + 7$ with the initial condition $a_1 = 4$ (the first week).

2. Finding the Explicit Formula:

The explicit formula for an arithmetic sequence is given by:

$a_n = a_1 + (n - 1) \cdot d$

Where:

• $a_1 = 4$ (the first week).
• $d = 7$ (the common difference).

Substituting these values:

$a_n = 4 + (n - 1) \cdot 7$

Simplifying:

$a_n = 4 + 7n - 7$

$a_n = 7n - 3$

So, the explicit formula is:

$a_n = 7n - 3$

3. Finding the Number of Cars Sold in the 4th Week:

To find the number of cars sold in the 4th week, we substitute $n = 4$ into the explicit formula:

$a_4 = 7(4) - 3$

$a_4 = 28 - 3 = 25$

So, 25 cars will be sold in the 4th week.

Final Answers:

• Recursive Formula: $a_n = a_{n-1} + 7$, with $a_1 = 4$.
• Explicit Formula: $a_n = 7n - 3$.
• Cars sold in the 4th week: 25 cars.

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Would you like any further details or clarifications?

Here are some related questions for further exploration:

1. How can you modify the recursive formula if the growth pattern were quadratic instead of linear?
2. What would happen if the dealership sold 4 cars in the first week but only sold 5 cars in the second week?
3. How can we interpret the explicit formula in real-world terms?
4. If the dealership sold 4 cars in the first week and then started decreasing sales, how would the recursive formula change?
5. How would the number of cars sold change in week 10 according to the explicit formula?

Tip: Always ensure to calculate the common difference first in an arithmetic sequence problem, as it plays a critical role in both the recursive and explicit formulas.