An arithmetic sequence is a sequence where the difference between any two consecutive terms is constant. This constant difference is called the common difference, and it allows the sequence to be generated by adding the same value to each term.
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The common difference in an arithmetic sequence can be positive, negative, or zero, but it must be constant throughout the sequence.
The explicit formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_n$ is the nth term, $a_1$ is the first term, and $d$ is the common difference.
The recursive formula for an arithmetic sequence is $a_n = a_{n-1} + d$, where $a_n$ is the nth term and $a_{n-1}$ is the previous term.
Arithmetic sequences can be used to model real-world situations, such as the number of dollars saved each month or the number of miles driven each day.
The sum of the first $n$ terms of an arithmetic sequence can be calculated using the formula $S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum, $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term.
Review Questions
Explain how the common difference affects the behavior of an arithmetic sequence.
The common difference in an arithmetic sequence determines the rate of change between consecutive terms. A positive common difference means the sequence is increasing, a negative common difference means the sequence is decreasing, and a common difference of zero means the sequence is constant. The magnitude of the common difference affects the rate of change, with larger common differences resulting in more rapid increases or decreases 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 any term in the sequence directly using its position $n$, the first term $a_1$, and the common difference $d$. In contrast, the recursive formula, $a_n = a_{n-1} + d$, allows you to calculate any term in the sequence by using the previous term $a_{n-1}$ and adding the common difference $d$. The two formulas are related, as the explicit formula can be derived from the recursive formula by repeatedly applying the recursive formula to find the nth term.
Analyze how the sum of the first $n$ terms of an arithmetic sequence can be used to model real-world situations.
The formula for the sum of the first $n$ terms of an arithmetic sequence, $S_n = \frac{n}{2}(a_1 + a_n)$, can be used to model a variety of real-world situations. For example, it can be used to calculate the total amount of money saved over a certain number of months, given the initial amount saved and the constant monthly savings. It can also be used to calculate the total distance traveled over a certain number of days, given the initial distance and the constant daily distance traveled. By understanding the relationship between the sum formula and the characteristics of an arithmetic sequence, you can apply this knowledge to analyze and solve problems involving real-world scenarios.