Problem 61 - 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

So, there are a total of 1430 seats in the theater.

Step 1: Identify the sequence

Firstly, determine that the problem describes an arithmetic progression. An arithmetic progression is a sequence of numbers in which the difference of any two successive members is a constant. In this case, the progression starts with 30 (the number of seats in the first row) and increases by 2 (the number of additional seats in each subsequent row).

Step 2: Identify the formula for sum of an arithmetic sequence

The formula for the sum 'S' of an arithmetic sequence is given by \(S = n/2 * (a + l)\), where 'n' is the number of terms (in this case, the number of rows, 26), 'a' is the first term (the number of seats in the first row, 30) and 'l' is the last term. The last term (number of seats in the last row) could be calculated by first term + (n - 1) * d, where 'd' is the common difference (the additional number of seats in each row, 2)

Step 3: Substitute in the values

Substitute the values into the formula: \(S = 26/2 * (30 + (30+(26-1)*2))\). To calculate the number of seats in the last row, we've used the formula: last term = first term + (n - 1) * difference

Step 4: Simplify the formula

By simplifying the expression, we obtain: \(S = 13 * (30 + 80) = 13 * 110\)

Step 5: Calculate the sum

Calculate the sum: \(S = 13 * 110 = 1430\) seats in total.