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Answer :
To determine the number of cars sold by the dealership in the sixth week, we need to use the explicit formula for the number of cars sold each week.
### Step-by-Step Solution
1. Understanding the Initial Problem:
- The dealership uses a linear growth model to track car sales. This means the number of cars sold increases by a constant amount each week.
- From the problem, we know:
- In the first week, 3 cars were sold (`P_0 = 3`).
- In the second week, 5 cars were sold (`P_1 = 5`).
- This shows that the number of cars sold increases by 2 cars each week.
2. Recursive Formula:
- The recursive formula provided is [tex]\( P_n = P_{n-1} + 2 \)[/tex]. This shows that each week, the dealership sells 2 more cars than the previous week.
3. Explicit Formula:
- The explicit formula given is [tex]\( P_n = 2 + 3n \)[/tex]. This formula calculates the number of cars sold in any [tex]\( (n+1) \)[/tex]th week directly, without needing information from previous weeks.
- In this formula, `n` represents the week number, starting from 0.
4. Calculate the Number of Cars Sold in the Sixth Week:
- For the sixth week, [tex]\( n+1 = 6 \)[/tex], which means [tex]\( n = 5 \)[/tex] since [tex]\( n \)[/tex] starts from 0.
- Substitute [tex]\( n = 5 \)[/tex] into the explicit formula:
[tex]\[
P_5 = 2 + 3 \times 5 = 2 + 15 = 17
\][/tex]
Therefore, 17 cars will be sold in the sixth week according to this linear growth model.
### Step-by-Step Solution
1. Understanding the Initial Problem:
- The dealership uses a linear growth model to track car sales. This means the number of cars sold increases by a constant amount each week.
- From the problem, we know:
- In the first week, 3 cars were sold (`P_0 = 3`).
- In the second week, 5 cars were sold (`P_1 = 5`).
- This shows that the number of cars sold increases by 2 cars each week.
2. Recursive Formula:
- The recursive formula provided is [tex]\( P_n = P_{n-1} + 2 \)[/tex]. This shows that each week, the dealership sells 2 more cars than the previous week.
3. Explicit Formula:
- The explicit formula given is [tex]\( P_n = 2 + 3n \)[/tex]. This formula calculates the number of cars sold in any [tex]\( (n+1) \)[/tex]th week directly, without needing information from previous weeks.
- In this formula, `n` represents the week number, starting from 0.
4. Calculate the Number of Cars Sold in the Sixth Week:
- For the sixth week, [tex]\( n+1 = 6 \)[/tex], which means [tex]\( n = 5 \)[/tex] since [tex]\( n \)[/tex] starts from 0.
- Substitute [tex]\( n = 5 \)[/tex] into the explicit formula:
[tex]\[
P_5 = 2 + 3 \times 5 = 2 + 15 = 17
\][/tex]
Therefore, 17 cars will be sold in the sixth week according to this linear growth model.
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