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 twelve cars ($P_1 = 12$).

1. Write the recursive formula for the number of cars sold.
2. Write the explicit formula for the number of cars sold, $P_n$, in the $(n+1)$th week.
3. If this trend continues, how many cars will be sold in the fourth week?

The recursive formula for the number of cars sold is Pn = Pn-1 + 7. The explicit formula for the number of cars sold is Pn= 5+7n. There will be 26 cars sold in the fourth week.

Explanation:

To write the recursive formula for the number of cars sold, we need to find the common difference between consecutive terms. The first week, the dealership sold 5 cars, and the second week it sold 12 cars. The common difference is found by subtracting the first term from the second term, which gives us 12 - 5 = 7. Therefore, the recursive formula is Pn = Pn-1 + 7.

To write the explicit formula for the number of cars sold, we can use the initial condition (P0 = 5) to find an equation that relates the week number (n) to the number of cars sold (Pn). The explicit formula is given by Pn = 5 + 7n.

To find the number of cars sold in the fourth week, we substitute n = 3 into the explicit formula: P3 = 5 + 7(3) = 5 + 21 = 26. Therefore, 26 cars will be sold in the fourth week.

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