College

We appreciate your visit to The number of cars sold weekly by a new automobile dealership grows according to a linear growth model The first week the dealership sold three. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

The number of cars sold weekly by a new automobile dealership grows according to a linear growth model. The first week the dealership sold three cars [tex]\((P_0 = 3)\)[/tex]. The second week the dealership sold five cars [tex]\((P_1 = 5)\)[/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 sixth week?
[tex]\[
\square \text{ cars}
\][/tex]

Answer :

Let's break down the problem step-by-step to find the recursive formula, the explicit formula, and the number of cars sold in the sixth week.

### Recursive Formula

1. Start with Given Information:
- In the first week, 3 cars were sold: [tex]\( P_0 = 3 \)[/tex].
- In the second week, 5 cars were sold: [tex]\( P_1 = 5 \)[/tex].

2. Determine the Pattern:
- The number of cars sold increased by 2 cars from the first week to the second week (5 - 3 = 2).

3. Write the Recursive Formula:
- Since the number of cars sold grows by 2 each week, the recursive formula is:
[tex]\[
P_n = P_{n-1} + 2
\][/tex]
- This means that each week, you add 2 to the number of cars sold in the previous week.

### Explicit Formula

1. Identify the Pattern in the Explicit Formula:
- The pattern increases linearly, starting at 3 cars in the first week.

2. Find the Explicit Formula:
- The formula can be written as:
[tex]\[
P_n = P_0 + (n - 1) \times 2
\][/tex]
- Here, [tex]\( P_0 = 3 \)[/tex] is the initial number of cars sold and [tex]\( (n - 1) \times 2 \)[/tex] accounts for the increase of 2 cars per week.

### Calculate Cars Sold in the Sixth Week

1. Use the Explicit Formula:
- To find the number of cars sold in the sixth week ([tex]\( n = 6 \)[/tex]), plug 6 into the explicit formula:
[tex]\[
P_6 = 3 + (6 - 1) \times 2 = 3 + 5 \times 2 = 3 + 10 = 13
\][/tex]
- So, the dealership will sell 13 cars in the sixth week.

### Summary

- Recursive Formula: [tex]\( P_n = P_{n-1} + 2 \)[/tex]
- Explicit Formula: [tex]\( P_n = 3 + (n - 1) \times 2 \)[/tex]
- Cars Sold in Sixth Week: 13 cars

Thanks for taking the time to read The number of cars sold weekly by a new automobile dealership grows according to a linear growth model The first week the dealership sold three. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada