Problem 92 - Precalculus with Limits

Question

Determine the seating capacity of an auditorium with 36 rows of seats if there are 15 seats in the first row, 18 seats in the second row, 21 seats in the third row, and so on.

Step-by-Step Solution

Answer

The seating capacity of the auditorium is 2430 seats.

Step 1: Identify the parameters of the arithmetic progression

The first term \(a=15\) in the sequence (the number of seats in the first row), the common difference \(d=3\) (each row has three more seats than the previous one), and the total number of terms \(n=36\) (the total number of rows).

Step 2: Use the summation formula of an arithmetic progression

The sum \(S\) of an arithmetic progression can be computed using the formula \(S =\frac{n}{2}(2a+(n-1)d)\). Subsequently, this will be used to calculate the total number of seats in the auditorium. Plug the already known values into the formula: \(S = \frac{36}{2}(2\times15+(36-1)\times3)\).

Step 3: Compute the sum

Simplify and solve the expression from the previous step: \(S = 18\times(30+105) = 18\times135 = 2430 \).

Key Concepts

Seating Capacity Calculation

Calculating seating capacity, especially in a structured setting like an auditorium, often involves sequencing and patterns. The seating capacity problem becomes much clearer when we recognize a pattern in the number of seats per row. In the given exercise, the number of seats increases by a constant amount with each new row, a pattern known as an 'arithmetic progression.' This is a common real-world application of arithmetic sequences, emphasizing the pattern's usefulness beyond the classroom.

When faced with this type of challenge, the goal is to estimate the total number of seats. To accomplish this, it’s essential to identify the starting point or the first term of the sequence, the pattern of how the sequence progresses, and how many terms (in this case, rows) there are in total. Once these elements are clear, the rest of the problem becomes simpler to tackle using arithmetic concepts.

Arithmetic Series

An arithmetic series, where we’re adding up the terms of an arithmetic progression, enables us to sum multiple values efficiently. In our exercise, the series begins with 15 seats in the first row and increases by 3 seats with every subsequent row. It’s important to know that an arithmetic series is used when you have a sequence of numbers with a common difference between them.

In a real-life scenario like seating in an auditorium, an arithmetic series reflects an organized increase in seats, perhaps due to design elements or construction constraints. Understanding how to apply the arithmetic series concept can greatly simplify calculations for seating arrangements, construction planning, resource distribution, and more. It’s a considerably powerful tool for any problem involving a regular interval of change.

Summation Formula

The summation formula for an arithmetic series is like a mathematical shortcut, it allows us to quickly find the sum of all terms within the sequence without having to add each term individually. Specifically, for an arithmetic series, the formula is given as
\( S = \frac{n}{2}(2a + (n - 1)d) \),
where
\( S \) is the sum of the series,
\( n \) is the number of terms,
\( a \) is the first term, and
\( d \) is the common difference between the terms.

This formula is derived from the concept of pairing terms equidistant from the beginning and the end of the series, thus simplifying the summation. For students and professionals alike, grasping this formula means saving time and effort, making it an essential tool in fields related to logistics, event planning, and finance, where summarizing a series of values is a frequent task.