Problem 70 - Algebra and Trigonometry

Question

A theater has 30 seats in the first row, 32 seats in the second row, increasing by 2 seats per row for a total of 26 rows. How many seats are there in the theater?

Step-by-Step Solution

Answer

The total number of seats in the theater is 1430.

Step 1: Identify the first term, common difference, and number of terms

The first term \( a_1 \) is 30. The common difference \( d \) is an increase of 2 seats per row. The number of terms \( n \), which corresponds to the number of rows, is 26.

Step 2: Calculate the last term

Calculate the last term \( a_n = a_1 + (n - 1) \cdot d \). The number of seats in the last row is \( 30 + (26 - 1)\cdot 2 = 30 + 50 = 80 \) seats.

Step 3: Calculate the sum of all terms

Finally, calculate the total number of seats using the formula for arithmetic sequence sum \( S_n = \frac{n}{2}(a_1 + a_n) \). The total number of seats is \( \frac{26}{2} \cdot (30 + 80) = 13 \cdot 110 = 1430 \) seats.