A theater has 29 rows of seats. The first row has 18 seats, the second row has 22 seats, the third row has 26 seats, and so on. How many se The theater has seats.
A theater has 29 rows of seats. The first row has 18 seats, the second row has 22 seats, the third row has 26 seats, and so on. How many se The theater has seats.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![**How to Determine Total Theater Seating**
A theater has 29 rows of seats. The first row has 18 seats, the second row has 22 seats, the third row has 26 seats, and so on. How many seats are in the theater in total?
The number of seats in the theater can be calculated using the information provided for the pattern in seat arrangements. Let's analyze the problem step-by-step:
1. **Understanding the Problem**:
- The theater has 29 rows.
- Each row increases the number of seats by a certain amount.
2. **Identifying the Pattern**:
- 1st row: 18 seats
- 2nd row: 22 seats
- 3rd row: 26 seats
- The number of seats increases by 4 seats for each subsequent row.
3. **Forming an Equation**:
Using the pattern, we can establish a general form for the number of seats in the nth row:
\[
a_n = 18 + (n-1) \cdot 4
\]
Where \( a_n \) represents the number of seats in the nth row.
4. **Summing Up**:
To find the total number of seats in the theater, we need to sum up the seats from row 1 to row 29. This can be done using the formula for the sum of an arithmetic series:
\[
S_n = \frac{n}{2} (a + l)
\]
Where \( S_n \) is the sum of the first n terms, \( a \) is the first term, and \( l \) is the last term.
Here:
- \( n = 29 \)
- \( a = 18 \)
- \( l = 18 + 28 \cdot 4 = 18 + 112 = 130 \)
Substituting the values, we get:
\[
S_{29} = \frac{29}{2}(18 + 130) = \frac{29}{2} \times 148 = 29 \times 74 = 2146
\]
5. **Conclusion**:
The total number of seats in the theater is 2146 seats.
**Solution**:
\[
\text{The theater has } \boxed{2146} \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b6c3136-3b78-4c3b-95a7-be26ea6fb45d%2Fafc5df0b-922b-4c69-9f3b-6b49a3ce3e6f%2Fwm3d6v7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**How to Determine Total Theater Seating**
A theater has 29 rows of seats. The first row has 18 seats, the second row has 22 seats, the third row has 26 seats, and so on. How many seats are in the theater in total?
The number of seats in the theater can be calculated using the information provided for the pattern in seat arrangements. Let's analyze the problem step-by-step:
1. **Understanding the Problem**:
- The theater has 29 rows.
- Each row increases the number of seats by a certain amount.
2. **Identifying the Pattern**:
- 1st row: 18 seats
- 2nd row: 22 seats
- 3rd row: 26 seats
- The number of seats increases by 4 seats for each subsequent row.
3. **Forming an Equation**:
Using the pattern, we can establish a general form for the number of seats in the nth row:
\[
a_n = 18 + (n-1) \cdot 4
\]
Where \( a_n \) represents the number of seats in the nth row.
4. **Summing Up**:
To find the total number of seats in the theater, we need to sum up the seats from row 1 to row 29. This can be done using the formula for the sum of an arithmetic series:
\[
S_n = \frac{n}{2} (a + l)
\]
Where \( S_n \) is the sum of the first n terms, \( a \) is the first term, and \( l \) is the last term.
Here:
- \( n = 29 \)
- \( a = 18 \)
- \( l = 18 + 28 \cdot 4 = 18 + 112 = 130 \)
Substituting the values, we get:
\[
S_{29} = \frac{29}{2}(18 + 130) = \frac{29}{2} \times 148 = 29 \times 74 = 2146
\]
5. **Conclusion**:
The total number of seats in the theater is 2146 seats.
**Solution**:
\[
\text{The theater has } \boxed{2146} \
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education