Understanding the Seating Arrangement

Oct 13, 2025 by recoveryshake.com 138 views

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Introduction

In this problem, we are given a movie auditorium with a specific seating arrangement. The first row has 45 seats, and there are 34 rows in total. Each successive row has two additional seats compared to the previous row. We need to find the total number of seats in the auditorium.

Understanding the Seating Arrangement

Let's break down the seating arrangement to understand the pattern. The first row has 45 seats, and each successive row has two additional seats. This means that the second row will have 45 + 2 = 47 seats, the third row will have 47 + 2 = 49 seats, and so on.

Calculating the Number of Seats in Each Row

To calculate the number of seats in each row, we can use the formula:

Number of seats in each row = Number of seats in the previous row + 2

We can start with the first row and apply this formula to find the number of seats in each subsequent row.

Row Number	Number of Seats
1	45
2	45 + 2 = 47
3	47 + 2 = 49
4	49 + 2 = 51
5	51 + 2 = 53
...	...

Finding the Total Number of Seats

To find the total number of seats in the auditorium, we need to sum up the number of seats in each row. However, since the number of seats in each row is increasing by 2, we can use the formula for the sum of an arithmetic series:

Sum = (n/2) * (a + l)

where n is the number of terms, a is the first term, and l is the last term.

In this case, the first term (a) is 45, and the last term (l) is 45 + 2 * (34 - 1) = 91. The number of terms (n) is 34.

Calculating the Sum

Now, let's plug in the values into the formula:

Sum = (34/2) * (45 + 91) Sum = 17 * 136 Sum = 2312

Conclusion

Therefore, the total number of seats in the movie auditorium is 2312.

Alternative Solution

We can also solve this problem using a different approach. We can start by finding the number of seats in the last row, which is 45 + 2 * (34 - 1) = 91. Then, we can use the formula for the sum of an arithmetic series:

Sum = (n/2) * (a + l)

where n is the number of terms, a is the first term, and l is the last term.

In this case, the first term (a) is 45, and the last term (l) is 91. The number of terms (n) is 34.

Sum = (34/2) * (45 + 91) Sum = 17 * 136 Sum = 2312

Final Answer

The total number of seats in the movie auditorium is 2312.

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Introduction

In our previous article, we solved a problem involving a movie auditorium with a specific seating arrangement. The first row has 45 seats, and there are 34 rows in total. Each successive row has two additional seats compared to the previous row. We found that the total number of seats in the auditorium is 2312.

Q&A

Q: What is the seating arrangement in the movie auditorium?

A: The seating arrangement in the movie auditorium is such that each successive row has two additional seats compared to the previous row.

Q: How many seats are in the first row?

A: There are 45 seats in the first row.

Q: How many rows are in the auditorium?

A: There are 34 rows in the auditorium.

Q: How many seats are in the last row?

A: The last row has 45 + 2 * (34 - 1) = 91 seats.

Q: How did you calculate the total number of seats in the auditorium?

A: We used the formula for the sum of an arithmetic series: Sum = (n/2) * (a + l), where n is the number of terms, a is the first term, and l is the last term.

Q: What is the total number of seats in the auditorium?

A: The total number of seats in the auditorium is 2312.

Q: Can you explain the formula for the sum of an arithmetic series?

A: The formula for the sum of an arithmetic series is: Sum = (n/2) * (a + l), where n is the number of terms, a is the first term, and l is the last term.

Q: What is the significance of the number of terms (n) in the formula?

A: The number of terms (n) represents the number of rows in the auditorium.

Q: Can you provide an alternative solution to the problem?

A: Yes, we can start by finding the number of seats in the last row, which is 45 + 2 * (34 - 1) = 91. Then, we can use the formula for the sum of an arithmetic series.

Q: What is the first term (a) in the formula?

A: The first term (a) is 45, which represents the number of seats in the first row.

Q: What is the last term (l) in the formula?

A: The last term (l) is 91, which represents the number of seats in the last row.

Q: Can you provide a step-by-step solution to the problem?

A: Here is a step-by-step solution to the problem:

Find the number of seats in the last row: 45 + 2 * (34 - 1) = 91.
Use the formula for the sum of an arithmetic series: Sum = (n/2) * (a + l), where n is the number of terms, a is the first term, and l is the last term.
Plug in the values into the formula: Sum = (34/2) * (45 + 91) = 2312.

Conclusion

We hope this Q&A article has provided a clear understanding of the problem and its solution. If you have any further questions or concerns, please don't hesitate to ask.