Question

The Jefferson Theatre contains 2508 seats. If the first row has 12 seats and each successive row has 4 additional seats than the previous row, determine the number of rows in the theatre.

A. 18
B. 20
C. 22
D. 24

Answer 1

To find the number of rows in the theatre, we can set up an arithmetic sequence using the number of seats in each row. By summing up the terms of the sequence, we can determine the total number of seats in the theatre. Solving for the number of rows gives us approximately 22.

Step-by-step explanation:

To determine the number of rows in the theatre, we need to find the arithmetic sequence that represents the number of seats in each row. Since the first row has 12 seats and each successive row has 4 additional seats than the previous row, we can represent this sequence as:

12, 16, 20, 24, ...

To find the total number of seats in the theatre, we can sum up the terms of this arithmetic sequence using the formula for the sum of an arithmetic series:

S = (n/2)(2a + (n-1)d)

where S is the sum, n is the number of terms, a is the first term, and d is the common difference.

In this case, a = 12 and d = 4. The total number of seats in the theatre is 2508. Substitute these values into the formula:

2508 = (n/2)(2(12) + (n-1)(4))

Now we can solve for n:

1. 2508 = (n/2)(24 + 4n - 4)
2. 2508 = (n/2)(20 + 4n)
3. 5016 = n(20 + 4n)
4. 5016 = 20n + 4n^2
5. 4n^2 + 20n - 5016 = 0

This is a quadratic equation. Solve for n using factoring, completing the square, or the quadratic formula. The solutions for n are approximately -23.13 and 21.63.

Since the number of rows cannot be negative, we discard the negative solution. Hence, the number of rows in the theatre is approximately 22.