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Answer :
To solve the problem of determining how many cars will be sold in the fifth week, let's first clarify both the recursive and explicit formulas, and then we can calculate the result step by step.
### Understanding the Problem
The car dealership sells cars following a linear growth model, meaning the number of cars sold increases by a consistent amount each week.
1. Recursive Formula:
The number of cars sold each week is given by the previous week's sales plus a fixed increase. The recursive formula is:
[tex]\[
P_n = P_{n-1} + 9
\][/tex]
This means that each week, the dealership sells 9 more cars than the previous week.
2. Explicit Formula:
The explicit formula provides the number of cars sold in terms of [tex]\( n \)[/tex], the number of weeks after the first week:
[tex]\[
P_n = 4 + 9n
\][/tex]
Here:
- [tex]\( 4 \)[/tex] is the number of cars sold in the first week.
- [tex]\( 9n \)[/tex] accounts for the increase of 9 cars each subsequent week.
### Calculating Cars Sold in the Fifth Week:
We need to find out how many cars are sold in the fifth week. Since the first week is [tex]\( n=0 \)[/tex], the fifth week corresponds to [tex]\( n=4 \)[/tex].
Using the explicit formula:
[tex]\[
P_4 = 4 + 9 \times 4
\][/tex]
Now, compute the number of cars sold:
[tex]\[
P_4 = 4 + 36 = 40
\][/tex]
Thus, the dealership will sell 40 cars in the fifth week.
### Understanding the Problem
The car dealership sells cars following a linear growth model, meaning the number of cars sold increases by a consistent amount each week.
1. Recursive Formula:
The number of cars sold each week is given by the previous week's sales plus a fixed increase. The recursive formula is:
[tex]\[
P_n = P_{n-1} + 9
\][/tex]
This means that each week, the dealership sells 9 more cars than the previous week.
2. Explicit Formula:
The explicit formula provides the number of cars sold in terms of [tex]\( n \)[/tex], the number of weeks after the first week:
[tex]\[
P_n = 4 + 9n
\][/tex]
Here:
- [tex]\( 4 \)[/tex] is the number of cars sold in the first week.
- [tex]\( 9n \)[/tex] accounts for the increase of 9 cars each subsequent week.
### Calculating Cars Sold in the Fifth Week:
We need to find out how many cars are sold in the fifth week. Since the first week is [tex]\( n=0 \)[/tex], the fifth week corresponds to [tex]\( n=4 \)[/tex].
Using the explicit formula:
[tex]\[
P_4 = 4 + 9 \times 4
\][/tex]
Now, compute the number of cars sold:
[tex]\[
P_4 = 4 + 36 = 40
\][/tex]
Thus, the dealership will sell 40 cars in the fifth week.
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