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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 a recursive formula for the sequence, we need to find the common difference between the number of seats in each row. The common difference can be found by subtracting the number of seats in the first row from the number of seats in the second row:
11 - 9 = 2
Therefore, the common difference is 2.
1. Recursive Formula:
Let a₁ be the number of seats in the first row (9 seats) and d be the common difference (2 seats). The recursive formula for the number of seats in each row is:
aₙ = aₙ₋₁ + d
2. Explicit Formula:
To find an explicit formula, we need to find a general formula for the nth term in the sequence. Since the first row has 9 seats and the common difference is 2, the explicit formula for the nth term can be written as:
aₙ = 9 + 2(n - 1)
aₙ = 9 + 2n - 2
aₙ = 2n + 7
3. Number of seats in the 12th row:
To find the number of seats in the 12th row, substitute n = 12 into the explicit formula:
a₁₂ = 2(12) + 7
a₁₂ = 24 + 7
a₁₂ = 31
Therefore, there are 31 seats in the 12th row.