A amphitheater has 15 seats in the first row, 18 in the second row, 21 in the third, and so on, for 30 rows. How many seats are in the amphitheater?

The number of seats in each row of the amphitheater follows an arithmetic sequence, where:

- The first row has 15 seats,
- The second row has 18 seats,
- The third row has 21 seats, and so on.

We can observe that the number of seats increases by 3 for each row. So, the common difference \( d = 3 \).

The formula for the \(n\)-th term of an arithmetic sequence is:

\[
a_n = a_1 + (n - 1) \cdot d
\]

Where:
- \( a_n \) is the number of seats in the \(n\)-th row,
- \( a_1 = 15 \) (the number of seats in the first row),
- \( d = 3 \) (the common difference),
- \( n \) is the row number.

To find the total number of seats in the amphitheater, we need to sum the number of seats across all 30 rows. The sum \(S_n\) of an arithmetic sequence can be found using the formula:

\[
S_n = \frac{n}{2} \cdot (a_1 + a_n)
\]

Where:
- \( S_n \) is the total number of seats,
- \( n = 30 \) (the number of rows),
- \( a_1 = 15 \),
- \( a_n \) is the number of seats in the 30th row.

First, let's find \(a_{30}\):

\[
a_{30} = 15 + (30 - 1) \cdot 3 = 15 + 87 = 102
\]

Now, use the sum formula:

\[
S_{30} = \frac{30}{2} \cdot (15 + 102) = 15 \cdot 117 = 1755
\]

Thus, the amphitheater has **1,755 seats**.

Would you like more details or further explanation?

Here are 5 related questions to expand on the concept:
1. How would the total number of seats change if the difference between rows increased?
2. What if there were 40 rows with the same arithmetic pattern?
3. How can you derive the formula for the sum of an arithmetic sequence?
4. What if the sequence was geometric instead of arithmetic?
5. How can this concept be applied to real-world seating designs?

**Tip:** Recognizing sequences as arithmetic (constant difference) or geometric (constant ratio) can help quickly solve many real-life problems!

Learn how to calculate the total number of seats in an amphitheater with 30 rows, where each row follows an arithmetic sequence. This problem demonstrates key concepts of arithmetic sequences, including finding the n-th term and summing the sequence. Suitable for Grades 9-11 students.

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