To find the total number of seats in the auditorium, we need to calculate the number of seats in each row and then sum them up.
The number of seats in each row can be expressed as follows:
- Row 1: 14 seats
- Row 2: 14 + 3 = 17 seats
- Row 3: 17 + 3 = 20 seats
- Row 4: 20 + 3 = 23 seats
- ...
We can see that the number of seats in row \( n \) can be described by the formula:
\[ \text{Seats in row } n = 14 + 3(n - 1) \]
So, the number of seats in each row is:
- Row 1: \( 14 + 3(1 - 1) = 14 \)
- Row 2: \( 14 + 3(2 - 1) = 17 \)
- Row 3: \( 14 + 3(3 - 1) = 20 \)
- Row 4: \( 14 + 3(4 - 1) = 23 \)
- ...
- Row 23: \( 14 + 3(23 - 1) = 14 + 66 = 80 \)
Thus, the number of seats in row \( n \) is \( 14 + 3(n - 1) \).
Now, we need to calculate the total number of seats from row 1 to row 23. The rows form an arithmetic sequence where:
- First term \( a = 14 \)
- Last term \( l = 80 \)
- Number of terms \( n = 23 \)
The sum \( S \) of an arithmetic sequence can be calculated using the formula:
\[ S = \frac{n}{2} \times (a + l) \]
Substituting the values we have:
\[ S = \frac{23}{2} \times (14 + 80) \] \[ S = \frac{23}{2} \times 94 \] \[ S = 23 \times 47 \] \[ S = 1081 \]
Therefore, the total number of seats in the 23 rows of Euclid High School's auditorium is 1081.
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