We are dealing with an arithmetic sequence where the number of seats in each row increases by 2 seats as we move from one row to the next.

Problem Breakdown:

• First row: 30 seats
• Second row: 32 seats
• Third row: 34 seats
• And so on...

We are given:

• First term of the sequence $a_1 = 30$
• Common difference $d = 2$
• Number of rows $n = 38$

To find the total number of seats, we need to calculate the sum of the first 38 terms of this arithmetic sequence. The formula for the sum of the first $n$ terms of an arithmetic sequence is:

$S_n = \frac{n}{2} \times (a_1 + a_n)$

Where:

• $S_n$ is the sum of the first $n$ terms
• $a_1$ is the first term
• $a_n$ is the $n$-th term

Step 1: Find the 38th term ($a_{38}$).

The formula for the $n$-th term of an arithmetic sequence is:

$a_n = a_1 + (n - 1) \times d$

Substitute the known values to find $a_{38}$:

$a_{38} = 30 + (38 - 1) \times 2 = 30 + 37 \times 2 = 30 + 74 = 104$

So, the 38th row has 104 seats.

Step 2: Calculate the total number of seats.

Now, use the sum formula:

$S_{38} = \frac{38}{2} \times (30 + 104) = 19 \times 134 = 2546$

Final Answer:

The amphitheater can seat 2,546 people.

Would you like further explanation or have additional questions? Here are some related questions:

1. How would the total number of seats change if the number of seats increased by 3 each row instead of 2?
2. How would you calculate the sum of the seats if there were 50 rows instead of 38?
3. How can we generalize the formula for the total number of seats if the row increment changes?
4. What is the 20th term (number of seats in the 20th row) in this amphitheater's seating pattern?
5. How would the total sum change if the first row had only 20 seats?

Tip: When solving problems with arithmetic sequences, it’s important to first identify the common difference and the formula for the $n$-th term, as these will be crucial in calculating sums and other terms efficiently.