Calculating the Total Number of Seats in a Theatre with an Arithmetic Sequence

In this article, we will explore how to calculate the total number of seats in a theatre when the seating pattern follows an arithmetic sequence. We will use the given example where the first row has 27 seats, the second row has 34 seats, and so forth, with the last row having 181 seats.

Understanding the Arithmetic Sequence

First, let's understand the pattern of the seating arrangement. We observe that: The first term (a1) is 27. The second term is 34, which is 27 7. The third term is 41, which is 34 7. This pattern continues until the last term, which is 181.

Determining the Number of Rows

To find the number of rows and the total number of seats, we need to determine the last term of the sequence (181) and the common difference (d 7).

Using the formula for the n-th term of an arithmetic sequence:

an a1 (n-1)d

Substituting the known values:

181 27 (n-1)7

Solving for (n-1):

181 - 27 7(n-1)

154 7(n-1)

n-1 22

n 23

Summing the Total Number of Seats

Now that we know there are 23 rows, we can use the sum formula for an arithmetic sequence to find the total number of seats.

Sn frac{n}{2} (a1 an)

S23 frac{23}{2} (27 181)

S23 frac{23}{2} (208)

S23 23 × 104

S23 2392

Alternative Methods

Here are a few alternative methods to verify the solution:

Directly using the sequence formula:

S23 frac{23}{2} (27 181) 2392

Using the sequence properties:

Total rows 181 - 27/7 22 1 23

Total seats 27 34 41 ... 181 2392

Another way of calculating:

92 (for first and 26th row) 2 × 46 (middle rows) 2392

Conclusion

To summarize, by understanding and applying the arithmetic sequence, we can determine the total number of seats in the theatre. For the given example, the theatre has 23 rows, and the total number of seats is 2392.

Using the formula and understanding the pattern, we can easily derive the solution for a similar problem.