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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 for the sequence, we first need to identify the common difference. Since the number of seats increases by 2 from one row to the next, the common difference is 2.
Recursive formula:
a(1) = 9
a(n) = a(n-1) + 2
To find the explicit formula for the sequence, we know the formula for the nth term of an arithmetic sequence is given by:
a(n) = a(1) + (n-1)d
Substitute a(1) = 9 and d = 2 into the formula:
a(n) = 9 + 2(n-1)
a(n) = 9 + 2n - 2
a(n) = 2n + 7
To find how many seats are in the 12th row, substitute 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.