Problem

A theater has 28 rows of seats. The first row has 16 seats, the second row has 18 seats, the third row has 20 seats, and so on. How many seats are in the theater?
The theatre has seats.

Answer

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Answer

Final Answer: The theater has \(\boxed{1204}\) seats.

Steps

Step 1 :This problem involves a theater with 28 rows of seats. The first row has 16 seats, the second row has 18 seats, the third row has 20 seats, and so on. We are asked to find the total number of seats in the theater.

Step 2 :This is an arithmetic sequence problem. The number of seats in each row forms an arithmetic sequence where the first term (\(a_1\)) is 16 and the common difference (\(d\)) is 2. The number of terms (\(n\)) is the number of rows, which is 28.

Step 3 :The sum (\(S\)) of an arithmetic sequence can be calculated using the formula: \(S = \frac{n}{2} * (2*a_1 + (n-1)*d)\)

Step 4 :We can substitute the given values into this formula to find the total number of seats in the theater. \(a_1 = 16\), \(d = 2\), and \(n = 28\).

Step 5 :Substituting these values into the formula, we get \(S = \frac{28}{2} * (2*16 + (28-1)*2)\)

Step 6 :Solving this expression, we find that \(S = 1204.0\)

Step 7 :Final Answer: The theater has \(\boxed{1204}\) seats.

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