Problem 79 - Beginning and Intermediate Algebra

Question

The main floor of a concert hall seats 860 people. The first row contains 24 seats, and the last row contains 62 seats. If each row has 2 more seats than the previous row, how many rows of seats are on the main floor of the concert hall?

Step-by-Step Solution

Answer

There are \(20\) rows of seats on the main floor of the concert hall.

Step 1: Identify the arithmetic sequence formula

For this exercise, we will use the arithmetic series formula: Sum = (n * (a1 + an))/2 Here: - Sum represents the total number of seats - n is the total number of rows - a1 is the number of seats in the first row - an is the number of seats in the last row We are given: - Sum = 860 - a1 = 24 - an = 62 We have to find the value of n.

Step 2: Plug in the given values into the formula

Now, we plug the given values into the arithmetic series formula: 860 = (n * (24 + 62))/2

Step 3: Simplify the equation and solve for n

Simplify the equation to solve for n: 860 = (n * 86)/2 Multiply both sides by 2 to get rid of the fraction: 1720 = n * 86 Now divide by 86 to isolate n: n = 1720 / 86 n = 20 So there are 20 rows of seats on the main floor of the concert hall.

Key Concepts

Arithmetic Series Formula

The arithmetic series formula is a crucial mathematical tool used for calculating the sum of terms in an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant is known as the "common difference."

The formula for the sum of the first n terms of an arithmetic sequence is:

• Sum = \( \frac{n}{2} (a_1 + a_n) \)

Where:

• Sum is the total sum of all terms in the sequence
• \( n \) is the number of terms
• \( a_1 \) is the first term
• \( a_n \) is the last term

This formula is particularly useful in situations where finding the exact number of terms or computing the sum manually would be tedious. Understanding this formula can help solve a variety of algebraic problems, such as seating arrangements or computing sums of sequences.

Concert Hall Seating

In this problem, the seating of a concert hall is structured in an arithmetic sequence. The first row in this setting holds a specific number of seats, and as you move to subsequent rows, the number of seats increases by a fixed number, which in this case is two. This structured increase is what makes the sequence arithmetic.

The first row has 24 seats, and as you go further back, each row gains 2 more seats than the previous one. By the time you reach the last row, it contains 62 seats. This orderly progression of seat numbers is typical in venues like concert halls where seating capacity adjustments are necessary to optimize viewing angles and access.

The understanding and application of arithmetic sequences in seating arrangements help planners efficiently utilize available space. This is done while making sure the audience has a comfortable and unobstructed view.

Number of Rows

Determining the number of rows in a situation like this involves using the arithmetic series formula. Knowing the total seating capacity, the number of seats in the first and last rows, you can compute the number of rows.

For our arithmetic series, it's given that the sum of seats is 860. With the first row having 24 seats and the last row having 62 seats, we substitute these into the formula for the sum of an arithmetic sequence.

From the equation \( 860 = \frac{n}{2} (24 + 62) \), simplifying gives us the number of rows \( n \). Solving this shows there are 20 rows. This process efficiently determines how many rows can fit into the concert hall while respecting the sequential seat increment.

Algebra Problem Solving

Solving algebra problems, especially with sequences and series, relies heavily on understanding how to manipulate equations. Key to this is identifying the correct formula to apply and correctly inserting the known values.

This exercise uses a systematic approach:

• Identify known variables
• Insert them into the formula
• Use algebra to solve for the unknown

For instance, we inserted the seating numbers into the arithmetic series formula to find the number of rows.

Simplifying the equation \( 860 = \frac{n \times 86}{2} \) involves eliminating fractions and isolating the variable \( n \).

• Multiply both sides by 2 to remove the fraction
• Divide by the sum of the starting and ending seats (86) to find \( n \)

This clear and ordered approach helps anyone facing algebraic problems to pinpoint solutions with confidence.