The numbers of seats in the first 12 rows of a high-school auditorium form an arithmetic sequence. The first row has 9 seats. The second row has 11 seats. Write a recursive formula to represent the sequence. Write an explicit formula to represent the sequence. How many seats are in the 12th row?
To find the recursive formula, we first need to find the common difference between each row's number of seats. The common difference is 11 - 9 = 2.
Therefore, the recursive formula to represent this sequence is:
a(n) = a(n-1) + 2, where a(1) = 9
To find the explicit formula, we can use the formula for arithmetic sequences:
a(n) = a(1) + (n-1)d, where a(1) = 9 and d = 2
So, the explicit formula to represent the sequence is:
a(n) = 9 + 2(n-1)
a(n) = 9 + 2n - 2
a(n) = 2n + 7
To find the number of seats in the 12th row, we can plug in n = 12 into the explicit formula:
a(12) = 2(12) + 7
a(12) = 24 + 7
a(12) = 31
Therefore, there are 31 seats in the 12th row of the auditorium.