Respuesta :
Answer:
A) Calculate [tex]P_{1}[/tex] and [tex]P_{2}[/tex].
For [tex]n=1[/tex]
[tex]P_{1} =P_{1-1} +30\\P_{1} =P_{0} +30=200+30=230[/tex]
[tex]\therefore P_{1}=230[/tex]
For [tex]n=2[/tex]
[tex]P_{2} =P_{2-1} +30\\P_{2} =P_{1} +30=230+30=260[/tex]
[tex]\therefore P_{2}=260[/tex]
B) Find explicit formula for [tex]P_{n}[/tex].
To find the explicit formula, you need the first term [tex]P_{0}=200[/tex] and the difference which is 30. Then, we use the arithmetic sequence definition
[tex]P_{n}= P_{0}+(n-1)d\\P_{n}=200+30(n-1) \\\therefore P_{n}=30n+170[/tex]
C) Use your formula to predict number of houses in 10 years.
For [tex]n=10[/tex]
[tex]P_{n}=30n+170\\P_{10}=30(10)+170=300+170=470[/tex]
In 10 years, the number of houses is 470.
D) When will the number of house reach 400.
For [tex]P_{n}=400[/tex], we have to find [tex]n[/tex]
[tex]P_{n}=30n+170\\400=30n+170\\30n=400-170\\n=\frac{230}{30} \approx 7.7[/tex]
After 7.7 years, approximately, the number of houses will reach 400.
A) P1 = 230
P2 = 260
B) An explicit formula for Pn is; Pn = 200 + 30n
C) The prediction of number of houses in 10 years is; 500 houses
D) The number of years it will take for the number of houses to reach 400 is; 7 years.
We are given;
Pn = P_(n - 1) + 30
Initial population; P_0 = 200
A) Since Pn = P_(n - 1) + 30, then;
P1 = P_(1 - 1) + 30
P1 = P_0 + 30
P1 = 200 + 30
P1 = 230
Similarly;
P2 = P1 + 30
P2 = 230 + 30
P2 = 260
B) Since P1 = 230 and P2 = 260, then the difference between 2 consecutive terms is; d = 260 - 230
d = 30
Thus;
Explicit formula for Pn is;
Pn = P0 + nd
Pn = 200 + 30n
C) From B above, we saw that;
Pn = 200 + 30n
Thus;
Number of houses in ten years is;
P(10) = 200 + 30(10)
P(10) = 500 houses
D) If number of houses is 400. Then;
400 = 200 + 30n
30n = 400 - 200
30n = 200
n = 200/30
n ≈ 7 years
Read more at; https://brainly.com/question/12374893
The complete question is;
The number of houses in a town has been growing according to the recursive rule Pn = P_(n - 1) + 30, with initial population P_0 = 200
A. calculate P1 and P2
B. Find an explicit formula for Pn
C. Use your formula to predict the number of houses in 10 years.
D. When will the number of houses reach 400 houses?
