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An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference. In the context of the theater problem, the sequence representing the number of seats in each row is an arithmetic sequence since each subsequent row has 3 more seats than the previous one. Here's how arithmetic sequences work:

Each term after the first is calculated by adding the common difference to the previous term. If the first term of the sequence is denoted by \( a \), the \( n \)-th term, also known as \( a_n \), can be expressed as:

\[ a_n = a + (n-1) imes d \]

where \( d \) is the common difference. For example, knowing that the first row has 20 seats (\( a = 20 \)) and each row adds 3 seats (\( d = 3 \)), the number of seats in the \( n \)-th row is calculated as \( a_n = 20 + (n-1) \times 3 \). This key concept helps determine the seat count for any row in the theater.

Finding the sum of an arithmetic series involves summing all terms in an arithmetic sequence. The sum of an arithmetic series can be calculated with a specific formula, particularly useful when dealing with a large number of terms, as in our theater problem.

The formula to find the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is:

\[ S_n = \frac{n}{2}(a + l) \]

where \( n \) is the total number of terms, \( a \) is the first term, and \( l \) is the last term. In the theater scenario, the first row has 20 seats, and the last row, as calculated, has 122 seats. With 35 rows to consider (thus \( n = 35 \)), using the formula gives:

\[ S_{35} = \frac{35}{2}(20 + 122) = 2485 \]

This means there are 2,485 seats in total in the theater. Understanding this formula is crucial when tasks involve summing large sets of sequential numbers methodically.

Word problems are a common way to test understanding of algebraic concepts, as they require translating real-world situations into mathematical expressions. Solving these problems generally involves several steps: interpreting the problem, setting up equations, and solving for the unknowns.

In the theater seating problem, translating the given information into a mathematical sequence and then finding the solution involves:

• Recognizing what is given and what needs to be found: the number of seats in each row and the total number of seats.
• Formulating the arithmetic sequence: starting with 20 seats and increasing by 3 each row.
• Using formulas for sequences and series to find the total number of seats.

Solving word problems requires a blend of reading comprehension and mathematical skills, making them excellent practice for applying algebra in practical situations.