Question

A special section in the end zone of a football stadium has 2 seats in the first row and 14 rows total. Each successive row has 2 seats more than the row before. In this particular section, the first seat is sold for 1 cent, and each following seat sells for \(5 \%\) more than the previous seat. Find the total revenue generated if every seat in the section is sold. Round only the final answer, and state the final answer in dollars rounded to two decimal places. (JJC) \(^{\dagger}\)

Short answer

The total revenue generated is approximately 3732.95 dollars.

Step-by-step solution

- Determine the number of seats in each row

The first row has 2 seats. Each subsequent row has 2 more seats than the previous row, forming an arithmetic sequence. The number of seats in the nth row is given by the formula: \[ a_n = a_1 + (n-1)d \] where \(a_1 = 2\) and \(d = 2\). The number of seats in the nth row is: \[ a_n = 2 + (n-1) \times 2 = 2n \]

- Calculate the total number of seats

Each row's number of seats can be calculated from Step 1. For 14 rows, sum the seats: \[ S = \sum_{n=1}^{14} 2n \] Sum of the first 14 terms of the sequence gives: \[ S = 2(1 + 2 + 3 + ... + 14) \]\[ = 2 \left(\frac{14 \times 15}{2}\right) \]\[ = 2 \times 105 \]\[ = 210 \]

- Determine the price of each seat

Starting at 1 cent, with each subsequent seat costing 5% more, form a geometric sequence. First term \(a = 0.01\) and common ratio \(r = 1.05\). The nth term in the geometric sequence is given by: \[ P_n = P_1 \times r^{(n-1)} \] For the 210th seat, the nth term calculates the price: \[ P_{210} = 0.01 \times 1.05^{209} \]

- Find total revenue

Sum the geometric sequence for 210 seats: \[ S_n = a \frac{1-r^n}{1-r} \] Replace values with \( a = 0.01, r = 1.05\), and \(n = 210\): \[ S = 0.01 \frac{1-1.05^{210}}{1 - 1.05} \] Calculate to find the total revenue.

- Compute and round

Evaluate the expression from Step 4 and round the result to the nearest cent. Using a calculator: \[ S \approx 3732.95 \] dollars.

Key concepts

arithmetic sequence

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is called the common difference and is denoted as \(d\). For example, consider the sequence: \(2, 4, 6, 8, \dots\). Here, each number is obtained by adding 2 (the common difference) to the previous number.

In the given exercise, the number of seats in each row forms an arithmetic sequence. The first row has 2 seats, and each subsequent row has 2 more seats than the previous row. This means \(a_1 = 2\) and \(d = 2\). To find the number of seats in the \^th row, use the formula:
\[ a_n = a_1 + (n-1)d \] So for any \ \ value, you substitute into this formula to get the number of seats in that particular row.

geometric sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio, denoted as \(r\). For example, consider the sequence: \(1, 2, 4, 8, \dots\). Each number is obtained by multiplying the previous number by 2.

In the stadium exercise, the price of each seat forms a geometric sequence. The first seat costs 1 cent, and each subsequent seat costs 5% more than the previous seat. This means the first term \(a = 0.01\) and the common ratio \(r = 1.05\). To find the price of the \^th seat, use the formula:
\[P_n = P_1 \times r^{(n-1)} \] By substiting into this formula, you can calculate the price of any specific seat in this sequence.

sum of series

The sum of a series is the addition of all terms in a sequence. For arithmetic sequences, the sum of the first \ \ terms can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (a_1 + a_n) \] where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the \^th term.
In our exercise, we calculated the total number of seats in the 14 rows using:
\[ S = 2 \times \frac{14 \times 15}{2} = 210 \]
For geometric sequences, the sum of the first \ \ terms is given by:
\[ S_n = a \frac{1-r^n}{1-r} \] where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
In our exercise, we used this to calculate the total revenue from 210 seats priced in a geometric sequence:
\[ S \times 0.01 = \frac{1 - 1.05^{210}}{1 - 1.05} \] This gave us a final value of approximately 3732.95 dollars.