Problem 29 - Algebra 2 and Trigonometry

Question

In a theater, there are 20 seats in the first row. Each row has 3 more seats than the row ahead of it. There are 35 rows in the theater. a. Express the number of seats in the \(n\) th row of the theater in terms of \(n .\) b. Use sigma notation to represent the number of seats in the theater.

Step-by-Step Solution

Answer

a. The nth row has \(3n + 17\) seats. b. Total seats: \(\sum_{n=1}^{35} (3n + 17) = 2485\).

Step 1: Understanding the Pattern

The first row has 20 seats. Each subsequent row has 3 more seats than the previous row. If the first row has 20 seats, the second row has 23, the third row has 26, and so on, we observe an arithmetic sequence with the first term 20 and a common difference of 3.

Step 2: Finding the nth Term of the Sequence

The nth term of an arithmetic sequence can be found using the formula for the nth term, which is given by \( a_n = a_1 + (n-1) \cdot d \). Here, \( a_1 = 20 \) and \( d = 3 \). Therefore, \( a_n = 20 + (n-1) \cdot 3 \). Simplify to get \( a_n = 3n + 17 \).

Step 3: Expressing Seats in nth Row

From the formula \( a_n = 3n + 17 \), we express the number of seats in the nth row as \( 3n + 17 \). This is the expression for part (a).

Step 4: Using Sigma Notation

To find the total number of seats in the theater, we need to sum the number of seats from row 1 to row 35. We use Sigma notation to represent this sum: \( \sum_{n=1}^{35} (3n + 17) \).

Step 5: Calculating the Total Number of Seats

The sum \( \sum_{n=1}^{35} (3n + 17) \) can be expanded as \( \sum_{n=1}^{35} 3n + \sum_{n=1}^{35} 17 \). Calculate the first part as \( 3 \sum_{n=1}^{35} n \), where \( \sum_{n=1}^{35} n = \frac{35 \times 36}{2} = 630 \), so the first sum is \( 3 \times 630 = 1890 \). The second sum is \( 17 \times 35 = 595 \). The total number of seats is \( 1890 + 595 = 2485 \).

Key Concepts

Sigma Notation

Sigma notation is a concise way to represent the sum of a sequence of numbers, and it's particularly useful in arithmetic sequences. In the context of our theater seating problem, we're interested in summing up the number of seats from row 1 to row 35.

The sigma notation is expressed as \( \sum \), followed by the expression to sum and the index of summation (often denoted by \( n \)). For example, if you wanted to sum the expression \( 3n + 17 \) from \( n = 1 \) to \( n = 35 \), you'd write it as \( \sum_{n=1}^{35} (3n + 17) \). This reads as "the sum of \( 3n + 17 \) as \( n \) goes from 1 to 35."

Sigma notation can be broken down further:

• \( n = 1 \) is the starting value of the index \( n \).
• \( 35 \) is the ending value.
• The expression \( 3n + 17 \) describes the pattern of seats in each row.

This notation makes it easier to set up and calculate large sums efficiently.

Sum of Arithmetic Series

The sum of arithmetic series involves adding up all the terms in an arithmetic sequence. In our problem, the sequence of seats follows an arithmetic pattern starting at 20 seats and increasing by 3 seats per row.

To find the sum: first, recognize the series starts at 20 seats and is defined by the formula for the nth term, \( a_n = 3n + 17 \), which means every term increases by 3. The formula for the sum \( S_n \) of an arithmetic series where the first term is \( a_1 \) and the last term is \( a_n \) is given by:

• \( S_n = \frac{n}{2} (a_1 + a_n) \)

Here:

• \( n \) is the number of terms.
• \( a_1 = 20 \) is the first term.
• The 35th term \( a_{35} = 3 \times 35 + 17 = 122 \).

Apply the formula:

• \( S_{35} = \frac{35}{2} (20 + 122) = \frac{35}{2} \times 142 = 2485 \).

So, the total number of seats is 2,485.

Finding nth Term

Finding the nth term of an arithmetic sequence is crucial for understanding patterns of increase or decrease in a sequence. In our example, each row in the theater seats more people than the row in front by a constant amount of 3 seats.

The general formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n-1) \cdot d \). Here's how it applies:

• \( a_1 = 20 \), representing the seats in the first row.
• \( d = 3 \), the common difference indicating how much more each row has compared to the previous one.

Plug these values into the formula:

• \( a_n = 20 + (n-1) \cdot 3 \)
• Simplify to get \( a_n = 3n + 17 \).

This formula \( a_n = 3n + 17 \) helps us calculate the seats in any row \( n \). For example, for row 10, the number of seats would be \( 3 \times 10 + 17 = 47 \).

So by understanding this formula, you can determine the seating for any row in this arithmetic sequence.