Question 1

The number of cars sold weekly by a new automobile dealership grows according to a linear growth model. The first week the dealership sold five cars [tex]$\left(P_0=5\right)$[/tex]. The second week the dealership sold seven cars [tex]$\left(P_1=7\right)$[/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]

To solve this problem, let's break it down step by step:

### Part 1: Recursive Formula

1. Identify the pattern:
The number of cars sold increases by a constant amount each week.

Given:
- Week 1: [tex]$P_0 = 5$[/tex]
- Week 2: [tex]$P_1 = 7$[/tex]

2. Determine the common difference:
The difference between the sales in Week 2 and Week 1 is:
[tex]\[
7 - 5 = 2
\][/tex]
Therefore, each week, the number of cars sold increases by 2.

3. Formulate the recursive formula:
The recursive formula for the number of cars sold can be written as:
[tex]\[
P_n = P_{n-1} + 2
\][/tex]
Here, [tex]$ P_n $[/tex] represents the number of cars sold in the [tex]$ (n+1) $[/tex]-th week.

### Part 2: Explicit Formula

1. Identify the general form of the arithmetic sequence:
An arithmetic sequence can be described by the formula:
[tex]\[
P_n = a + (n-1)d
\][/tex]
where [tex]$ a $[/tex] is the first term and [tex]$ d $[/tex] is the common difference.

2. Substitute the values:
Given [tex]$ a = 5 $[/tex] (the first week's sales) and [tex]$ d = 2 $[/tex]:
[tex]\[
P_n = 5 + 2n
\][/tex]
This formula represents the number of cars sold in the [tex]$ (n+1) $[/tex]-th week.

### Part 3: Number of Cars Sold in the Fourth Week

1. Calculate the number of cars sold in the fourth week:
According to the explicit formula, for [tex]$ n = 3 $[/tex] (since we start counting from [tex]$ P_0 $[/tex]):
[tex]\[
P_3 = 5 + 2(3) = 5 + 6 = 11
\][/tex]

So, the dealership will sell 11 cars in the fourth week.

### Summary

1. Recursive Formula:
[tex]\[
P_n = P_{n-1} + 2
\][/tex]

2. Explicit Formula:
[tex]\[
P_n = 5 + 2n
\][/tex]

3. Number of cars sold in the fourth week:
[tex]\[
11 \text{ cars}
\][/tex]