A large multiplex movie house has many theaters.  The largest theater has 33 rows.  There are 17 seats in the first row.  Each row has two seats more than the previous row.  How many total seats are there in this​ theater?

To find the total number of seats in the largest theater, we need to calculate the number of seats in each of the 33 rows and then sum them all.

The number of seats in each row forms an arithmetic sequence where:
- The first term (\(a_1\)) is 17 (seats in the first row),
- The common difference (\(d\)) is 2 (because each row has 2 more seats than the previous row),
- The number of terms (\(n\)) is 33 (because there are 33 rows).

The formula for the sum 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,
- \(a_n\) is the last term.

### Step 1: Find the last term \(a_n\)
The general formula for the \(n\)-th term of an arithmetic sequence is:

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

Substitute the values:

\[
a_{33} = 17 + (33 - 1) \times 2 = 17 + 64 = 81
\]

So, the 33rd row has 81 seats.

### Step 2: Calculate the sum of the seats
Now, use the sum formula for an arithmetic sequence:

\[
S_{33} = \frac{33}{2} \times (17 + 81) = \frac{33}{2} \times 98 = 33 \times 49 = 1617
\]

Thus, the total number of seats in the theater is **1,617**.

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Would you like more details or further clarification on any steps?

Here are five related questions to further your understanding:
1. How would the total seats change if each row had three more seats than the previous one?
2. Can you generalize the formula to calculate the total seats for any number of rows and any seat increment per row?
3. How does this problem relate to the concept of arithmetic series in real life?
4. What if the rows decreased in number of seats instead of increasing? How would that affect the sum?
5. Could you solve the problem using a different method, such as summing each row manually?

**Tip**: Whenever solving an arithmetic series problem, remember to first determine both the first term and the common difference clearly.

A large multiplex movie house has many theaters. The largest theater has 33 rows. There are 17 seats in the first row. Each row has two seats more than the previous row. How many total seats are there in this theater?

Learn how to calculate the total number of seats in a theater using arithmetic sequence concepts. The theater has 33 rows, with 17 seats in the first row, and each subsequent row has 2 additional seats. Detailed step-by-step solution provided, suitable for Grades 9-10.

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