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Answer :
We begin by noting that the dealership’s sales are described by a linear growth model. This means that the number of cars sold increases by a constant amount each week. We are given:
- In the first week, the dealership sold
[tex]$$P_0=2$$[/tex]
- In the second week, it sold
[tex]$$P_1=11$$[/tex]
Step 1. Determine the weekly increase.
The constant increase (change in sales) is the difference between the second and the first week:
[tex]$$11-2=9$$[/tex]
Step 2. Write the recursive formula.
Since each week the previous week’s sales increase by 9 cars, the recursive relationship is:
[tex]$$P_n = P_{n-1} + 9.$$[/tex]
Step 3. Write the explicit formula.
An explicit formula for a linear growth process takes the form:
[tex]$$P_n = P_0 + 9n,$$[/tex]
and since [tex]$P_0=2$[/tex] we have:
[tex]$$P_n=2+9n.$$[/tex]
Step 4. Calculate the number of cars sold in the fourth week.
Using the explicit formula, when the trend is evaluated “in” the fourth week we get:
[tex]$$P_n=2+9n\quad\text{with } n=4.$$[/tex]
Thus, the number of cars sold in the fourth week is:
[tex]$$P_4=2+9(4)=36\text{ cars.}$$[/tex]
Step 5. Calculate the number of cars sold after the fourth week.
Continuing the trend, after the fourth week the number of cars sold is:
[tex]$$P_{\text{after 4th week}}=47\text{ cars.}$$[/tex]
For clarity, here are the final answers:
1. Recursive formula:
[tex]$$P_n=P_{n-1}+9.$$[/tex]
2. Explicit formula:
[tex]$$P_n=2+9n.$$[/tex]
3. Number of cars sold in the fourth week: 36 cars.
4. Number of cars sold after the fourth week: 47 cars.
- In the first week, the dealership sold
[tex]$$P_0=2$$[/tex]
- In the second week, it sold
[tex]$$P_1=11$$[/tex]
Step 1. Determine the weekly increase.
The constant increase (change in sales) is the difference between the second and the first week:
[tex]$$11-2=9$$[/tex]
Step 2. Write the recursive formula.
Since each week the previous week’s sales increase by 9 cars, the recursive relationship is:
[tex]$$P_n = P_{n-1} + 9.$$[/tex]
Step 3. Write the explicit formula.
An explicit formula for a linear growth process takes the form:
[tex]$$P_n = P_0 + 9n,$$[/tex]
and since [tex]$P_0=2$[/tex] we have:
[tex]$$P_n=2+9n.$$[/tex]
Step 4. Calculate the number of cars sold in the fourth week.
Using the explicit formula, when the trend is evaluated “in” the fourth week we get:
[tex]$$P_n=2+9n\quad\text{with } n=4.$$[/tex]
Thus, the number of cars sold in the fourth week is:
[tex]$$P_4=2+9(4)=36\text{ cars.}$$[/tex]
Step 5. Calculate the number of cars sold after the fourth week.
Continuing the trend, after the fourth week the number of cars sold is:
[tex]$$P_{\text{after 4th week}}=47\text{ cars.}$$[/tex]
For clarity, here are the final answers:
1. Recursive formula:
[tex]$$P_n=P_{n-1}+9.$$[/tex]
2. Explicit formula:
[tex]$$P_n=2+9n.$$[/tex]
3. Number of cars sold in the fourth week: 36 cars.
4. Number of cars sold after the fourth week: 47 cars.
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