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The number of cars sold weekly by a new automobile dealership grows according to a linear growth model. The first week the dealership sold eight cars \((P_0 = 8)\). The second week the dealership sold fourteen cars \((P_1 = 14)\).

1. Write the recursive formula for the number of cars sold, \(P_n\), in the \((n + 1)\)th week.
\[ P_n = P_{n-1} + \]

2. Write the explicit formula for the number of cars sold, \(P_n\), in the \((n + 1)\)th week.
\[ P_n = \]

3. If this trend continues, how many cars will be sold in the sixth week?
\[ \text{Cars:} \]

Answer :

The recursive formula for the number of cars sold is Pn = Pn-1 + 6. The explicit formula is Pn = 8 + 6n. In the sixth week, the dealership will sell 38 cars.

The problem involves finding a recursive formula and an explicit formula for the number of cars sold by a dealership, followed by a prediction for the number of cars sold in the sixth week.

To find the recursive formula, we observe the pattern from the given data: the dealership started with 8 cars in the first week (P0 = 8) and sold 14 cars in the second week (P1 = 14). The increase from the first week to the second week is 14 - 8 = 6 cars. Since this is a linear growth model, the number of cars sold increases by the same amount each week. Therefore, the recursive formula is Pn = Pn-1 + 6, where Pn represents the number of cars sold in the (n + 1)th week.

To derive the explicit formula, we use the initial value (8 cars sold in the first week) and the common difference (6 cars). The explicit formula is Pn = 8 + 6n.

For the sixth week (n = 5 as the first week is n = 0), we calculate P5 using the explicit formula: P5 = 8 + 6(5) = 38 cars will be sold.

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