In Euclid High School's auditorium, the first row contains 14 seats, the second row contains 17 seats, etc., adding 3 additional seats per row until the back row, row 23. How many total seats are in the 23 rows of Euclid High School's auditorium? (1 point)

Question

Responses

276
276

253
253

1081
1081

80
80

Answer 1

To find the total number of seats in the auditorium, we need to determine how many seats are in each row and then sum them up.

The first row has 14 seats. Each subsequent row adds 3 more seats than the previous row. Therefore, the number of seats in each row can be described as follows:

• Row 1: 14 seats
• Row 2: 14 + 3 = 17 seats
• Row 3: 17 + 3 = 20 seats
• Row 4: 20 + 3 = 23 seats
• ...
• Row \( n \): Seats in row 1 + \( (n-1) \times 3 \)

Thus, the formula for the number of seats in row \( n \) is: \[ \text{Seats in row } n = 14 + (n-1) \times 3 \]

We need to calculate the total number of seats from Row 1 to Row 23.

To find the number of seats in Row 23: \[ \text{Seats in row 23} = 14 + (23 - 1) \times 3 = 14 + 22 \times 3 = 14 + 66 = 80 \]

Now, we need to find the total number of seats in all 23 rows. This is the sum of the arithmetic series where:

• The first term \( a = 14 \)
• The last term \( l = 80 \)
• The number of terms \( n = 23 \)

The sum \( S_n \) of an arithmetic series can be calculated using the formula: \[ S_n = \frac{n}{2} (a + l) \]

Plugging in our values: \[ S_{23} = \frac{23}{2} (14 + 80) = \frac{23}{2} \times 94 = 23 \times 47 = 1081 \]

Thus, the total number of seats in the 23 rows of Euclid High School's auditorium is 1081.