An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, and it allows the terms of the sequence to be generated in a predictable pattern.
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The common difference in an arithmetic sequence can be positive, negative, or zero, which determines whether the sequence is increasing, decreasing, or constant, respectively.
The explicit formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_n$ is the $n$th term, $a_1$ is the first term, and $d$ is the common difference.
The recursive formula for an arithmetic sequence is $a_{n+1} = a_n + d$, where $a_{n+1}$ is the next term, $a_n$ is the current term, and $d$ is the common difference.
The sum of the first $n$ terms of an arithmetic sequence can be calculated using the formula $S_n = \frac{n}{2}[2a_1 + (n-1)d]$, where $S_n$ is the sum, $a_1$ is the first term, and $d$ is the common difference.
Arithmetic sequences have many real-world applications, such as in finance (e.g., calculating interest payments), physics (e.g., motion with constant acceleration), and computer science (e.g., loop structures).
Review Questions
Explain how the common difference affects the behavior of an arithmetic sequence.
The common difference in an arithmetic sequence determines whether the sequence is increasing, decreasing, or constant. If the common difference is positive, the sequence is increasing; if the common difference is negative, the sequence is decreasing; and if the common difference is zero, the sequence is constant. The magnitude of the common difference also affects the rate of change between consecutive terms in the sequence.
Describe the relationship between the explicit formula and the recursive formula for an arithmetic sequence.
The explicit formula for an arithmetic sequence, $a_n = a_1 + (n-1)d$, allows you to calculate the value of any term in the sequence based on its position $n$ and the first term $a_1$ and common difference $d$. The recursive formula, $a_{n+1} = a_n + d$, defines each term in the sequence based on the previous term and the common difference. While the explicit formula provides a direct way to find any term, the recursive formula describes the pattern of how the terms are generated, which can be useful for understanding the sequence's behavior and generating the terms iteratively.
Analyze how the sum of the first $n$ terms of an arithmetic sequence is related to the sequence's common difference and the first and last terms.
The formula for the sum of the first $n$ terms of an arithmetic sequence, $S_n = \frac{n}{2}[2a_1 + (n-1)d]$, demonstrates the relationship between the sum, the common difference $d$, and the first term $a_1$. The common difference $d$ affects the rate of change in the sequence, which in turn influences the overall sum. Additionally, the first term $a_1$ and the number of terms $n$ are both factors in the formula, indicating that the sum is dependent on the starting point and the length of the sequence. By understanding this relationship, you can use the sum formula to solve problems involving arithmetic sequences, such as calculating the total distance traveled in a motion problem or the total interest earned in a financial application.