Question: The number of cars sold weekly by a new automobile dealership grows according to a linear growth model. If this week the dealership sold seven cars ($P_0 = 7$) and next week the dealership sold nine cars ($P_1 = 9$) then:
(a) Write the recursive formula for the number of cars sold, $P_n$, in the $n^{th}$ week.
$P_n = P_{n-1} + $
(b) Write the explicit formula for the number of cars sold, $P_n$, in the $n^{th}$ week.
$P_n = $
(c) If this trend continues, how many cars will be sold in the sixth week?
cars
1. **State the problem:** We have a linear growth model for the number of cars sold weekly. Given $P_0 = 7$ and $P_1 = 9$, find:
(a) The recursive formula for $P_n$.
(b) The explicit formula for $P_n$.
(c) The number of cars sold in the sixth week, $P_6$.
2. **Find the common difference:** Since the growth is linear, the number of cars sold increases by a constant amount each week.
Calculate the difference:
$$d = P_1 - P_0 = 9 - 7 = 2$$
This means each week, the number of cars sold increases by 2.
3. **(a) Recursive formula:**
The recursive formula for a linear sequence is:
$$P_n = P_{n-1} + d$$
Substitute $d=2$:
$$P_n = P_{n-1} + 2$$
4. **(b) Explicit formula:**
The explicit formula for a linear sequence is:
$$P_n = P_0 + n \times d$$
Substitute $P_0=7$ and $d=2$:
$$P_n = 7 + 2n$$
5. **(c) Number of cars sold in the sixth week:**
Calculate $P_6$ using the explicit formula:
$$P_6 = 7 + 2 \times 6 = 7 + 12 = 19$$
**Final answers:**
(a) $P_n = P_{n-1} + 2$
(b) $P_n = 7 + 2n$
(c) $19$ cars will be sold in the sixth week.