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Answer :
(a) Recursive formula for the number of cars sold, [tex]P_n[/tex], in the (n + 1)th week: [tex]P_n = P_{n-1} + 2[/tex]
(b) Explicit formula for the number of cars sold, [tex]P_n[/tex], in the (n + 1)th week:
[tex]P_n = 6 + 2n[/tex]
(c) In the fourth week, the dealership will sell 12 cars.
To find the recursive formula for the number of cars sold, [tex]P_n[/tex], in the (n + 1)th week, we can observe the pattern from the given information.
Given data:
[tex]P_0 = 6[/tex] (number of cars sold in the first week)
[tex]P_1 = 8[/tex] (number of cars sold in the second week)
We can see that each week, the number of cars sold increases by 2. So, the recursive formula can be expressed as:
[tex]P_n = P_{n-1} + 2[/tex]
This formula states that the number of cars sold in the nth week [tex](P_n)[/tex] is equal to the number of cars sold in the previous week [tex](P_{n-1})[/tex] plus 2.
Next, let's find the explicit formula for the number of cars sold, [tex]P_n[/tex], in the (n + 1)th week.
To do this, we need to identify the initial value ([tex]P_0[/tex]) and the common difference (d) in the arithmetic sequence. In this case, the initial value is 6 (P₀ = 6), and the common difference is 2 (the number of cars sold increases by 2 each week).
The explicit formula for an arithmetic sequence is given by:
[tex]P_n = P_0 + n * d[/tex]
Substitute the given values:
[tex]P_n = 6 + n * 2[/tex]
Therefore, the explicit formula for the number of cars sold, P_n, in the (n + 1)th week is [tex]P_n = 6 + 2n[/tex].
Now, let's find how many cars will be sold in the fourth week (n = 3):
[tex]P_3 = 6 + 2 * 3[/tex]
[tex]P_3 = 12[/tex]
In the fourth week, the dealership will sell 12 cars.
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Final answer:
The recursive formula for the number of cars sold weekly is Pn = Pn - 1 + 2, where Pn represents the number of cars in the nth week. The explicit formula is Pn = 6 + 2n. Based on this model, the dealership would sell 14 cars in the fourth week.
Explanation:
In your problem, the number of cars sold each week is increasing by a constant amount, following a linear growth model. The first week, 6 cars were sold and the second week, 8 cars were sold. This indicates an increase of 2 cars from week 1 to week 2.
To write the recursive formula for the number of cars sold, Pn, in the (n + 1)th week, we determine the growth by subtraction: P1 - P0 = 8 - 6 = 2. So, every week, the number of cars sold increases by 2. The recursive formula would therefore be Pn = Pn - 1 + 2.
For the explicit formula which gives us the number of cars sold in the n-th week directly, we observe that it started with 6 cars (base) and increments by 2 each week. Therefore, the explicit formula would be Pn = 6 + 2n.
By the fourth week, the number of cars sold would be P4 = 6 + 2 * 4 = 6 + 8 = 14 cars sold.
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