Calculating Seats in an Auditorium with an Arithmetic Series

Understanding the seating arrangement in an auditorium is crucial for planning events. In this article, we will explore how to use the concept of arithmetic progression to calculate the number of seats in the last row and the total number of seats in an auditorium with a specific seating arrangement.

Seating Arrangement: A Real-World Example

Consider an auditorium where the first row has 12 seats, and each successive row has 2 additional seats. If the auditorium has a total of 20 rows, how many seats are in the last row, and what is the total number of seats in the auditorium?

Seats in the Last Row

The number of seats in each row can be described using an arithmetic progression. Let's denote the number of seats in the n-th row as Sn.

The formula for the number of seats in the n-th row is given by:

Sn 12 2(n - 1)

We can derive this formula as follows:

The first row has 12 seats. Each subsequent row adds 2 more seats than the previous row.

To find the number of seats in the 20th row, we substitute n 20 into the formula:

S20 12 2(20 - 1) 12 2 × 19 12 38 50

Therefore, the last row (20th row) has 50 seats.

Total Number of Seats in the Auditorium

To find the total number of seats in all 20 rows, we need to sum the number of seats in each row from the first to the 20th row. The number of seats in each row forms an arithmetic sequence with the first term a 12, the common difference d 2, and the number of terms n 20.

The sum (Sn) of the first n terms of an arithmetic sequence is given by:

Sn (frac{n}{2}) × (2a (n - 1)d)

Substituting the values:

S20 (frac{20}{2}) × (2 × 12 (20 - 1) × 2) 10 × (24 38) 10 × 62 620

Therefore, the total number of seats in the auditorium is 620.

Conclusion

By using the arithmetic progression formula and the sum of an arithmetic series, we can accurately determine the number of seats in the last row and the total number of seats in the auditorium. For the given example, the last row has 50 seats, and the auditorium has a total of 620 seats.

Additional Notes

The problem of calculating the number of seats in the last row and the total number of seats also serves as a practical application of arithmetic sequences. Each row represents a term in the arithmetic sequence, with the first term and the common difference providing the necessary parameters to calculate the required values.

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