Problem 74 - Precalculus: Mathematics for Calculus

Question

A drive-in theater has spaces for 20 cars in the first parking row, 22 in the second, 24 in the third, and so on. If there are 21 rows in the theater, find the number of cars that can be parked.

Step-by-Step Solution

Answer

840 cars.

Step 1: Recognize the Pattern

The number of spaces in each row increases by 2. This suggests an arithmetic sequence, where the number of spaces starts at 20 in the first row and increases by 2 in each subsequent row.

Step 2: Identify the Sequence Components

Identify the first term \(a_1 = 20\) and the common difference \(d = 2\) for the arithmetic sequence. There are 21 rows, so \(n = 21\).

Step 3: Calculate the Last Term

Use the formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n-1) \, d\). Substitute \(a_1 = 20\), \(d = 2\), and \(n = 21\). Thus, \(a_{21} = 20 + (21-1) \, 2 = 20 + 40 = 60\).

Step 4: Calculate the Total Number of Cars

The formula for the sum of the first \(n\) terms of an arithmetic sequence is \(S_n = \frac{n}{2} \, (a_1 + a_n)\). Substitute \(n = 21\), \(a_1 = 20\), and \(a_{21} = 60\):\[S_{21} = \frac{21}{2} \, (20 + 60) = \frac{21}{2} \, 80 = 21 \, \times \, 40 = 840\].

Key Concepts

Understanding the Sum of Arithmetic Series

An arithmetic series is the total sum of the terms in an arithmetic sequence. In our example with the drive-in theater, the sum of spaces across the 21 rows forms an arithmetic series. The number of spaces increases in each row, creating a specific pattern, and our task is to find the total number of parking spaces available.

To find the total sum, you can use the formula for the sum of an arithmetic series:

• \( S_n = \frac{n}{2} \times (a_1 + a_n) \)

In this formula:- \( S_n \) is the sum of the sequence.- \( n \) is the number of terms.- \( a_1 \) is the first term of the sequence.- \( a_n \) is the last term of the sequence.

By substituting the values from our exercise \( n = 21 \), \( a_1 = 20 \), and \( a_{21} = 60 \), we can find the total number of cars that can be parked: \( S_{21} = \frac{21}{2} \times (20 + 60) = 840 \). This means 840 cars can be parked in all 21 rows.

Role of the Common Difference in Sequences

In an arithmetic sequence, the common difference is the fixed amount that every pair of consecutive terms differs by. For the drive-in theater, this common difference is what made each row of parking spaces larger by a consistent number.

Here, the common difference is 2. This means each subsequent row has exactly 2 more spaces than the previous row. It is a critical component in forming a regular and predictable sequence, essential when calculating terms and understanding the overall pattern of the sequence.

To find the common difference in any arithmetic sequence, just subtract the first term from the second:

• \( d = a_{2} - a_{1} \)

Using the common difference, you can efficiently compute current and subsequent terms of the sequence without having to start from scratch every time.

Using the nth Term Formula

The nth term formula is a powerful tool for finding a specific term's value in an arithmetic sequence without having to list all prior terms. In our example, to find the number of spaces in the 21st row (the last row), we used the nth term formula.

The formula is:

• \( a_n = a_1 + (n-1) \, d \)

Where:- \( a_n \) is the nth term.- \( a_1 \) is the first term of the sequence.- \( d \) is the common difference.- \( n \) is the term number you want to find.

In the theater problem, substituting \( a_1 = 20 \), \( d = 2 \), and \( n = 21 \), we get: \( a_{21} = 20 + (21-1) \, 2 = 60 \). Hence, the 21st row can hold 60 cars.

Sequence Components Explained

Understanding the components of an arithmetic sequence is crucial for proper calculation and prediction of terms. In any sequence, you have several key components:

• The first term \( a_1 \): This is your starting point. For the parking rows, it starts at 20 spaces.
• The common difference \( d \): It adds incrementally to the first term and subsequent terms. Here, the rows increase by 2 spaces each.
• The number of terms \( n \): This tells you about the length of the sequence, in this case, there are 21 rows.

These components help structure any arithmetic sequence and guide you in calculating other terms precisely through formulas. If you misidentify any component, it can lead to incorrect conclusions. It is these components that enable a straightforward calculation of the sum and terms of a sequence.