Problem 60 - Algebra and Trigonometry

Question

Drive-ln Theater 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

The theater can park 840 cars.

Step 1: Identify the Sequence

The number of cars in each row follows an arithmetic sequence. The first row has 20 cars, the second row has 22 cars, and the third row has 24 cars, increasing by 2 each row. This sequence follows the pattern: 20, 22, 24, ..., where each term increases by 2.

Step 2: Calculate the Last Term

To find the number of cars in the last row, we use the formula for the nth term of an arithmetic sequence: \(a_n = a + (n-1)d\), where \(a\) is the first term (20), \(n\) is the number of terms (21), and \(d\) is the common difference (2). Substituting these values, we get \(a_{21} = 20 + (21-1) \times 2 = 20 + 40 = 60\). So, the last row has 60 cars.

Step 3: Sum the Sequence

The sum of all the cars in the 21 rows can be found using the formula for the sum of an arithmetic sequence: \(S_n = \frac{n}{2} (a + l)\), where \(n\) is the number of terms (21), \(a\) is the first term (20), and \(l\) is the last term (60). Plugging in the values gives us \(S_{21} = \frac{21}{2} (20 + 60) = \frac{21}{2} \times 80 = 21 \times 40 = 840\).

Key Concepts

Arithmetic Sequence Formula

An arithmetic sequence is a series of numbers where each term increases by a constant difference. In our example, the number of cars increases by the same amount with each row. This is called the "common difference." To describe this sequence systematically, we use the arithmetic sequence formula:

• First term \(a\)
• Number of terms \(n\)
• Common difference \(d\)

The formula to find the n-th term of an arithmetic sequence is:\[a_n = a + (n-1) \times d\]This helps us find any term in the sequence without listing all of them.

Sum of an Arithmetic Sequence

To find how many cars fit in all 21 rows, we need the sum of the sequence. The sum formula for arithmetic sequences is useful here:\[S_n = \frac{n}{2} (a + l)\]where:

• \(S_n\) is the sum of the first \(n\) terms
• \(n\) is the number of terms
• \(a\) is the first term
• \(l\) is the last term

For our problem, substituting the known values gives us the total number of cars that can be parked in 21 rows.

Nth Term Formula

Finding the n-th term of an arithmetic sequence allows us to calculate how many cars are in any row directly. We use the formula:\[a_n = a + (n-1) \times d\]This is particularly helpful for finding the last term in the sequence, as it avoids calculating every previous term. For our scenario with 21 rows, this formula quickly gives us the number of cars in the final row, which is a crucial step to using sum formulas.

Common Difference

The constant number added to each term of the arithmetic sequence is the common difference. In the theater's parking problem, each row has two more cars than the previous row. Recognizing the common difference helps identify the arithmetic nature of the sequence, and is used across the arithmetic sequence formulas including both the nth term and the sum formulas.

Sequence Sum Calculation

Once we know the first term, the last term, and the number of terms, we can calculate the sum quickly. In our example, we first find the last term using the nth term formula. Then, apply:\[S_n = \frac{n}{2} (a + l)\]This simplifies finding the total sum of cars across all rows, which for 21 rows resulted in 840 cars. It showcases the elegance and utility of arithmetic sequence formulas in solving real-world problems.