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The number of cars sold weekly by a new automobile dealership grows according to a linear growth model. The first week, the dealership sold two cars [tex]\((P_0=2)\)[/tex]. The second week, the dealership sold ten cars [tex]\((P_1=10)\)[/tex].

1. Write the recursive formula for the number of cars sold, [tex]P_n[/tex], in the [tex](n+1)^{\text{th}}[/tex] week.

[tex]P_n = P_{n-1} + \square[/tex]

2. Write the explicit formula for the number of cars sold, [tex]P_n[/tex], in the [tex](n+1)^{\text{th}}[/tex] week.

[tex]P_n = \square[/tex]

3. If this trend continues, how many cars will be sold in the fourth week?

[tex]\square \text{ cars}[/tex]

Answer :

To solve the problem of finding the recursive and explicit formulas for the number of cars sold by the dealership, as well as determining how many cars will be sold in the fourth week, let's break it down step-by-step:

1. Identify the Pattern:
- In the first week, the dealership sold 2 cars.
- In the second week, they sold 10 cars.
- This shows a pattern of growth, and we need to find out how much the sales increase each week.

2. Determine the Common Difference:
- The difference in the number of cars sold from the first week to the second week is 10 - 2 = 8 cars.
- This common difference (8) represents the number of additional cars sold each subsequent week.

3. Write the Recursive Formula:
- A recursive formula expresses the number of cars sold in terms of the previous week.
- Given the pattern, we can write the recursive formula as:
[tex]\[
P_n = P_{n-1} + 8
\][/tex]
- This means each week, the dealership sells 8 more cars than the previous week.

4. Write the Explicit Formula:
- An explicit formula allows us to find the number of cars sold in any week without knowing the previous week's sales.
- Starting from the first week, the formula is:
[tex]\[
P_n = 2 + n \times 8
\][/tex]
- Here, "2" is the initial number of cars sold, and "n \times 8" accounts for the growth over n weeks.

5. Calculate Cars Sold in the Fourth Week:
- Using the explicit formula, we substitute [tex]\( n = 3 \)[/tex] because the fourth week corresponds to [tex]\( n+1 = 4 \)[/tex].
- Calculate [tex]\( P_3 = 2 + 3 \times 8 \)[/tex].
- This results in [tex]\( P_3 = 2 + 24 = 26 \)[/tex].

Thus, if the trend continues, 26 cars will be sold in the fourth week.

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