Problem 91
Question
Determine the seating capacity of an auditorium with 30 rows of seats if there are 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on.
Step-by-Step Solution
Verified Answer
The auditorium can seat a total of 2340 people.
1Step 1: Identify Arithmetic Progression
First, determine that the rows of seats follow an arithmetic progression pattern. Starting from the first row (which has 20 seats), each subsequent row adds 4 additional seats compared with the previous row (24 in the second, 28 in the third, and so on). This means that the common difference \(d\), which is the difference between each successive term, is 4.
2Step 2: Use Arithmetic Progression Sum Formula
Next, calculate the sum of the arithmetic progression. This sequence has a finite number of terms – 30 (because there are 30 rows). The formula for the sum \(S\) of an arithmetic progression is given by \(S = 0.5 * n * (a + l)\), where \(n\) is the number of terms, \(a\) is the first term and \(l\) is the last term.
3Step 3: Calculate the Last Term
To use the formula stated above, the last term \(l\) of the progression needs to be found. The formula for any term in an arithmetic progression is \(l = a + (n - 1) * d\), where \(n\) is the number of terms, \(a\) is the first term and \(d\) is the common difference. Substituting \(a\) = 20, \(d\) = 4 and \(n\) = 30 into the equation results in \(l = 20 + (30 - 1) * 4 = 136\).
4Step 4: Find the Sum (Total Capacity)
Substitute the values found into the sum formula as follows: \(S = 0.5 * 30 * (20 + 136) = 0.5 * 30 * 156 = 2340\).
Key Concepts
Arithmetic SequenceArithmetic SeriesSum of Arithmetic Series
Arithmetic Sequence
Understanding arithmetic sequences is vital for solving many problems, just like the seating capacity of an auditorium. Imagine a sequence of numbers where the difference between consecutive terms is always the same. This constant difference is what defines an arithmetic sequence.
For example, if you start with 20 seats in the first row and keep adding 4 seats for each subsequent row, you form an arithmetic sequence: 20, 24, 28, and so on. These numbers illustrate a very structured pattern, where each number is obtained simply by adding the common difference (in this case, 4) to the previous term.
Applied to scenarios such as seating arrangements, heights of a stacked tower, or even dates on a calendar, arithmetic sequences appear often in both mathematical problems and real-world situations. Recognizing these patterns is the first step in breaking down many problems into more manageable pieces.
For example, if you start with 20 seats in the first row and keep adding 4 seats for each subsequent row, you form an arithmetic sequence: 20, 24, 28, and so on. These numbers illustrate a very structured pattern, where each number is obtained simply by adding the common difference (in this case, 4) to the previous term.
Applied to scenarios such as seating arrangements, heights of a stacked tower, or even dates on a calendar, arithmetic sequences appear often in both mathematical problems and real-world situations. Recognizing these patterns is the first step in breaking down many problems into more manageable pieces.
Arithmetic Series
Once we've recognized the sequence of numbers in tasks like determining total auditorium seating, we encounter an 'arithmetic series'. An arithmetic series is the sum of the terms of an arithmetic sequence. For instance, when adding up the number of seats across all rows in the auditorium, we aren't just looking at individual numbers but rather the total sum of those numbers.
This leads us to the concept of a series, where we accumulate values in a sequence to find an overall total. It's like summing up your total expenses over a month or calculating the total distance you've travelled after several consistent trips.
The beauty of dealing with an arithmetic series in particular is that there's an efficient way to find this total sum without having to add up every single term individually, which becomes particularly handy when dealing with a large number of terms.
This leads us to the concept of a series, where we accumulate values in a sequence to find an overall total. It's like summing up your total expenses over a month or calculating the total distance you've travelled after several consistent trips.
The beauty of dealing with an arithmetic series in particular is that there's an efficient way to find this total sum without having to add up every single term individually, which becomes particularly handy when dealing with a large number of terms.
Sum of Arithmetic Series
The 'sum of an arithmetic series' is incredibly useful when considering how to calculate the total number of something - such as seats in an auditorium. There's a simple formula to calculate this sum, and it depends on knowing just a few pieces of information: the total number of terms, the first term, and the last term of the series.
The general formula is given as \( S = n/2 (a + l) \), where \( S \) is the sum of the series, \( a \) is the first term, \( l \) is the last term, and \( n \) is the number of terms. It works because it essentially pairs up the first term with the last term, the second term with the second-to-last term, and so on, which results in a series of equal sums, making the arithmetic much simpler.
For our auditorium problem, we calculated the final term (the seats in the last row) and then used this formula to find the total number of seats. This method provides a fast and efficient way to sum a long sequence without direct addition of all terms, saving both time and effort when faced with extensive arithmetic series.
The general formula is given as \( S = n/2 (a + l) \), where \( S \) is the sum of the series, \( a \) is the first term, \( l \) is the last term, and \( n \) is the number of terms. It works because it essentially pairs up the first term with the last term, the second term with the second-to-last term, and so on, which results in a series of equal sums, making the arithmetic much simpler.
For our auditorium problem, we calculated the final term (the seats in the last row) and then used this formula to find the total number of seats. This method provides a fast and efficient way to sum a long sequence without direct addition of all terms, saving both time and effort when faced with extensive arithmetic series.
Other exercises in this chapter
Problem 90
In Exercises 89 and 90, consider a job offer with the given starting salary and the given annual raise. (a) Determine the salary during the sixth year of employ
View solution Problem 90
In Exercises 85-96, find the sum. \( \displaystyle \sum_{i=0}^{5} 3i^2 \)
View solution Problem 91
In Exercises 85-96, find the sum. \( \displaystyle \sum_{k=0}^{3} \frac{1}{k^2 + 1} \)
View solution Problem 92
In Exercises 91 - 94, prove the identity. \( _nC_n = _nC_0 \)
View solution