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Chapter 11: Problem 70

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?

Short Answer

Expert verified
The total number of seats in the theater is 1430.

Step by step solution

01

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.
02

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.
03

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.

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