Question

1. In an auditorium, there are 22 seats in the first row and 28 seats in the second row. The number of seats in a row continues to increase by 6 with each additional row.

(a) Write an iterative rule to model the sequence formed by the number of seats in each row. Show your work.
(b) Use the rule to determine which row has 100 seats. Show your work.

Answer 1

Explanation:

a) We can find the iterative rule by first making a table of values.

n ->
`a_n`

1 -> 22

2 -> 28

3 -> 34

We see that
`a_n` increases by 6 each time. Hence, the iterative rule should have the term 6n. However, if we put 1 in for n, we get 6. We want 22-6=16 more than this, or
`6n+16`. This would work for any seat row we give.

b) Our iterative rule to get the number of seats is
`6n+16`. Since we know that this value is 100, we can put both equal to each other.

`6n+16=100\\6n=84\\n=14`

Thus, the 14th row has 100 seats.

Answer 2

(a) S_n = S_1 + 6 (n - 1)

(b) Row 14 has 100 seats

Explanation:

(a) The arithmetic sequence would follow the iterative rule,

S_n = S_1 + 6 (n - 1) where S is the number of seats in row n, n is the row number, and S is the number of seats in row 1.

Row 1 = S_1 = 22

2 = S_2 = 28

3 = S_3 = 34

4 = S_4 = 40

|| || = 46

|| || = 52

|| || = 58

|| || = 64

|| || = 70

|| || = 76

|| || = 82

|| || = 88

|| || = 94

Row 14 = S_14 = 100

(b) 100=22+6(n-1) solve for n after setting S_n = 100

100 = 22 + 6_n - 6

84/6 = 6n/6 Row 14 has 100 seats

14 = n

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