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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 eight cars [tex]\((P_0=8)\)[/tex]. The second week, the dealership sold twelve cars [tex]\((P_1=12)\)[/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 4th week?

[tex]\(\square\)[/tex] cars

Answer :

Sure! Let's tackle this problem step-by-step.

The problem asks us to find the recursive and explicit formulas for the number of cars sold each week by a new automobile dealership. From the information given:

1. In the first week, the dealership sold 8 cars ([tex]\(P_0 = 8\)[/tex]).
2. In the second week, the dealership sold 12 cars ([tex]\(P_1 = 12\)[/tex]).

### 1. Recursive Formula:

A recursive formula expresses the quantity in terms of the previous term(s). For this linear growth model, we see that the number of cars sold is increasing by a constant amount each week.

- Step 1: Calculate the weekly increase:
[tex]\[
\text{Increase} = P_1 - P_0 = 12 - 8 = 4 \text{ cars}
\][/tex]
This tells us that each week, the dealership sells 4 more cars than the week before.

- Step 2: Write the recursive formula:
[tex]\[
P_n = P_{n-1} + 4
\][/tex]

### 2. Explicit Formula:

An explicit formula expresses the quantity directly in terms of [tex]\(n\)[/tex] (the week number, starting from 0).

- The explicit formula can be derived by recognizing that each week, the sales increase by 4 cars:
[tex]\[
P_n = P_0 + n \times \text{Increase}
\][/tex]

- Since [tex]\(P_0 = 8\)[/tex] and the increase is 4 cars per week, the explicit formula is:
[tex]\[
P_n = 8 + n \times 4
\][/tex]

### 3. Number of Cars Sold in 4th Week:

Now, we want to find out how many cars will be sold in the 4th week.

- The 4th week corresponds to [tex]\(n = 3\)[/tex] (since we start counting weeks from 0).

- Using the explicit formula:
[tex]\[
P_3 = 8 + 3 \times 4 = 8 + 12 = 20 \text{ cars}
\][/tex]

So, in the 4th week, the dealership will sell 20 cars.

In summary:
- The recursive formula is: [tex]\( P_n = P_{n-1} + 4 \)[/tex].
- The explicit formula is: [tex]\( P_n = 8 + n \times 4 \)[/tex].
- In the 4th week, the dealership sells 20 cars.

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