Understanding Arithmetic Sequences: Theatre Seating Arrangement

When we encounter problems involving sequences and series, particularly in practical scenarios such as theatre seating arrangements, we can apply mathematical concepts to find solutions. In this article, we will explore how an arithmetic sequence can be used to solve a real-world problem related to theatre seating. The problem involves determining which row in a theatre has 60 seats and calculating the total number of seats in the theatre.

Arithmetic Sequence Basics

An arithmetic sequence is a sequence of numbers in which the difference between any two successive members is constant. This constant difference is referred to as the 'common difference'

The Theatre Seating Problem

Determining the Row with 60 Seats

The seating arrangement in a mini-theatre is such that the first row has 8 seats, the second row has 12 seats, and the third row has 16 seats. The pattern continues, with each subsequent row containing 4 more seats than the previous one. We need to determine which row has 60 seats.

The general formula for the nth term of an arithmetic sequence is:

T_n a (n - 1)d

Where:

T_n is the nth term of the sequence, a is the first term of the sequence (8 in this case), d is the common difference (4 in this case), n is the term number.

Applying the Formula

Here's the step-by-step solution to find the row with 60 seats:

Given: 60 a (n - 1)d Substitute the values: 60 8 (n - 1)4 Simplify: 60 8 4n - 4 Further simplify: 60 4n 4 Solve for n: 56 4n Therefore, n 14

Hence, the 14th row has 60 seats.

Total Number of Seats in the Theatre

The theatre has 20 rows, and we need to calculate the total number of seats in the theatre. The formula for the sum of the first n terms of an arithmetic sequence is:

S_n

Given: n 20, a 8, d 4

Substitute the values: S_20 Simplify: S_20 10(16 76) Further simplify: S_20 10(92) Therefore, S_20 920

Hence, the total number of seats in the theatre is 920.

Conclusion

By using the properties of arithmetic sequences, we were able to determine that 60 seats are located in the 14th row. Additionally, we calculated the total number of seats in the theatre to be 920. This application of arithmetic sequences demonstrates the practical use of these concepts in real-world scenarios such as theatre seating and other similar problems.

Keywords: arithmetic sequence, theatre seating, common difference