Problem 70 - College Algebra

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 series parameters

The series starts with 30 seats, increments by 2 seats per row, and ends at the 26th row. So, this is an arithmetic series where the first term (a) is 30, the common difference (d) is 2, and the number of terms (n), which is the number of rows, is 26.

Step 2: Use the Arithmetic Series Formula

The formula for the sum (S) of an arithmetic series is given by \(S = \frac{n}{2} [2a + (n-1)d]\). Here, 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms.

Step 3: Substituting Values into the Formula

By substituting the values into the formula, we get \(S = \frac{26}{2} [2*30 + (26-1) * 2] = 13 * [60 + 50] = 13 * 110\)

Step 4: Calculate the Total Number of Seats

Next, we calculate the product to find the sum: \( S = 13 * 110 = 1430\). This represents the total number of seats in the theater.