Question: The number of cars sold weekly by a new automobile dealership grows according to a linear growth…

The number of cars sold weekly by a new automobile dealership grows according to a linear growth model. The first week the dealership sold five cars (\(P_0 = 5\)). The second week the dealership sold six cars (\(P_1 = 6\)).

Write the recursive formula for the number of cars sold, \(P_N\), in the \((N + 1)^{\text{th}}\) week.

\[ P_N = P_{N-1} + \boxed{\phantom{1}} \]

Write the explicit formula for the number of cars sold, \(P_N\), in the \((N + 1)^{\text{th}}\) week.

\[ P_N = \boxed{\phantom{1}}N + \boxed{\phantom{1}} \]

If this trend continues, how many cars will be sold in the fourth week?

\(\boxed{\phantom{1}}\) cars

In each box, enter a single number. Do not leave any boxes blank (enter 1 or 0 as appropriate if necessary).

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Solution

Let’s solve the problem step by step. Step 1: Determine the common difference in the linear growth model. \[ d = P_1 - P_0 = 6 - 5 = 1 & \] Step 2: Write the recursive formula using the common difference. \[ P_N = P_{N-1} + 1 & \] Step 3: Develop the explicit formula based on the linear growth model. \[ P_N = 1 \cdot N + 5 & \] Step 4: Calculate the number of cars to be sold in the fourth week. Since \( N = 3 \) corresponds to the fourth week: \[ P_3 = 1 \cdot 3 + 5 = 8 & \] Final Answer: 8