Question

Seats in an Amphitheater An outdoor amphitheater has 35 seats in the first row, 37 in the second row, 39 in the third row, and so on. There are 27 rows altogether. How many can the amphitheater seat?

Short answer

The amphitheater can seat 1647 people.

Step-by-step solution

- Identify the Sequence Type

Recognize that the number of seats in each row forms an arithmetic sequence because the number increases by a constant difference. The first term, denoted as \(a_1\), is 35. The common difference, \(d\), is 2.

- Determine the Number of Rows

The problem states there are 27 rows in total. This means \(n = 27\).

- Find the Number of Seats in the Last Row

Use the formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1) d\). Substituting the values: \[a_{27} = 35 + (27 - 1) \cdot 2 = 35 + 52 = 87\]. So, the 27th row has 87 seats.

- Use the Formula for Sum of an Arithmetic Series

The sum of the first \(n\) terms of an arithmetic sequence is given by: \(S_n = \frac{n}{2} (a_1 + a_n)\). Substituting the values: \[S_{27} = \frac{27}{2} (35 + 87)\]

- Calculate the Total Number of Seats

Perform the calculations: \[S_{27} = \frac{27}{2} \cdot 122 = 27 \cdot 61 = 1647\]. Therefore, the total number of seats in the amphitheater is 1647.

Key concepts

arithmetic sequence

An arithmetic sequence is a series of numbers where the difference between consecutive terms is always constant. This difference is known as the common difference and is typically denoted by the letter \(d\). For instance, in the provided exercise, the number of seats in each row of the amphitheater increases by 2, which is the common difference. The sequence starts with 35 seats (the first term denoted by \(a_1\)) and continues by adding the common difference to each term to get the next one.
So the sequence looks like this:
\[35, 37, 39, 41, \ldots\]
This regular pattern makes it easier to predict future terms or analyze data.

sum of arithmetic series

To find the total number of seats in the amphitheater, we need to calculate the sum of the arithmetic series.
The sum of an arithmetic series is the total when all terms in the sequence are added together. For calculation, we can use the formula:
\[S_n = \frac{n}{2}(a_1 + a_n)\]
where \(S_n\) represents the sum of the first \(n\) terms, \(a_1\) is the first term, and \(a_n\) is the nth term.
Using the formula, we find the last term of the sequence using:
\[a_n = a_1 + (n - 1) d\]
In this exercise, we have 27 rows (n = 27), and thus:\[a_1 = 35\] (first term) \(d = 2\) (common difference), resulting in the nth term:
\[a_{27} = 35 + (27 - 1) \cdot 2 = 87\]
Summing up, we substitute the values into the sum formula:
\[S_{27} =\frac{27}{2}\times(35 + 87)= 1647\]
Therefore, the total number of seats is 1647.

sequence and series

Before diving into solving problems, it's helpful to distinguish between a sequence and a series.
A sequence is an ordered list of numbers where each number is called a term. It can be finite or infinite. For example, the number of seats per row in the amphitheater forms a finite arithmetic sequence:
\[35, 37, 39, 41, \ldots, 87\]
On the other hand, a series is the sum of the terms in a sequence. When we add up all the terms in the sequence of seats, it becomes an arithmetic series. So, the exercise technically deals with an arithmetic series because it ultimately asks for the sum of all the rows of seats.
This distinction is crucial as it guides which formulas and methods to use for finding solutions.