Problem 62 - College Algebra

Question

A section in a stadium has 20 seats in the first row, 23 seats in the second row, increasing by 3 seats per row for a total of 38 rows How many seats are in this section of the stadium?

Step-by-Step Solution

Answer

The total number of seats in the section of the stadium is given by the sum of the arithmetic sequence, which can be calculated by substituting the correct values into the formula, simplifying and evaluating.

Step 1: Identify values

Identify the values necessary for the calculation. Here, the first term (a) is 20, the common difference (d) is 3 and the number of terms (n) is 38.

Step 2: Formula for the Sum of Arithmetic Progression

To find the sum (S) of an arithmetic sequence, one can use the formula \( S = \frac{n}{2} [2a + (n-1)d] \) where n is the number of terms, a is the first term and d is the common difference.

Step 3: Substitute values

Substitute the appropriate values into the formula. That would give \( S = \frac{38}{2} [2*20 + (38-1)*3] \)

Step 4: Simplify and calculate the sum

Simplify the expression inside the square brackets and then evaluate the multiplication and addition to find the sum. This should give the total number of seats in this section of the stadium.

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is referred to as the 'common difference' and is a defining characteristic of any arithmetic sequence.

For example, in the context of the stadium seating problem, starting with 20 seats in the first row and increasing by 3 seats per row creates an arithmetic progression. Here, each term after the first is formed by adding the common difference of 3 to the previous term. To visualize:

• 1st row: 20 seats
• 2nd row: 23 seats (20 + 3)
• 3rd row: 26 seats (23 + 3), and so on.

This sequence continues consistently throughout the 38 rows, making it possible to predict the number of seats in any row without having to count each one individually.

Arithmetic Sequence

An arithmetic sequence is essentially another term for an arithmetic progression. It represents a series of numbers with a specific, regular interval between them. In educational terms, it's important to emphasize that 'sequence' refers to the list of numbers, while each number in the list is called a 'term'.

When looking at our stadium seating problem, we called the number of seats in the first row the 'first term' of the sequence. It's denoted by the letter 'a'. The 'common difference' (d), which is the number of extra seats added to each row, helps us to find any term in the sequence. Using a simple formula, the nth term of an arithmetic sequence can be found by the expression: \( a_n = a + (n - 1)d \), where \( a_n \) is the nth term, \( n \) is the term's position in the sequence, and \( d \) is the common difference.

Sum Formula

The sum formula for an arithmetic sequence is used when we need to find the total of all terms up to a certain point. This is crucial for problems like calculating the total number of seats across multiple rows of a stadium.

To find the sum, the formula \( S = \frac{n}{2} [2a + (n-1)d] \) is used.

• \( S \) stands for the sum of the series.
• \( n \) is the number of terms being added.
• \( a \) is the first term in the series.
• \( d \) is the common difference.

In our stadium example, by substituting in these values—\( n = 38 \), \( a = 20 \), and \( d = 3 \)—you will find the total number of seats in this section of the stadium. This formula is incredibly useful in calculating the sum without the need to individually add up each term of the sequence, saving time and reducing potential for error.