Mathematical sequences can be used to model real life applications. Suppose you want to construct a movie theater in your town. The number of seats in each row can be modeled by the formula C_n = 16 + 4n, when n refers to the nth row, and you need 50 rows of seats.
(a) Write the sequence for the number of seats for the first 5 rows
(b) How many seats will be in the last row?
(c) What will be the total number of seats in the theater?
a.
C(n)=16+4n
C(1)=16+4(1)=20
C(2)=16+4(2)=24
...
C(5)=16+4(5)=36
b.
C(50)=16+4(50)=?
c.
50
∑ C(i)
i=1
50
∑ 16+4i
i=1
16(50)+4sum(i)
=800 + 4[ 50(1+50)/2 ]
= 800+4(1275)
= 5900
To answer these questions, we need to understand the given formula and use it to calculate the number of seats in each row.
(a) The formula C_n = 16 + 4n represents the number of seats in the nth row. Substituting different values of n, we can find the number of seats in the first five rows:
For n = 1:
C_1 = 16 + 4(1) = 16 + 4 = 20
For n = 2:
C_2 = 16 + 4(2) = 16 + 8 = 24
For n = 3:
C_3 = 16 + 4(3) = 16 + 12 = 28
For n = 4:
C_4 = 16 + 4(4) = 16 + 16 = 32
For n = 5:
C_5 = 16 + 4(5) = 16 + 20 = 36
Therefore, the sequence for the number of seats in the first five rows is 20, 24, 28, 32, 36.
(b) To find the number of seats in the last row, we use the formula with n = 50 (since there are 50 rows in total):
C_50 = 16 + 4(50) = 16 + 200 = 216
Therefore, the last row will have 216 seats.
(c) To find the total number of seats in the theater, we sum up the number of seats in all 50 rows:
Total number of seats = C_1 + C_2 + C_3 + ... + C_50
To simplify this calculation, we can use the formula for the sum of an arithmetic series:
Sum = (first term + last term) * number of terms / 2
In this case, the first term is C_1 = 20, the last term is C_50 = 216, and the number of terms is 50:
Total number of seats = (20 + 216) * 50 / 2 = 236 * 25 = 5900
Therefore, the total number of seats in the theater will be 5900.