The common difference is a term used in the context of sequences, which are ordered lists of numbers or objects that follow a specific pattern. The common difference refers to the constant value by which each term in the sequence differs from the previous term.
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The common difference is the constant value that is added to each term in an arithmetic sequence to get the next term.
In an arithmetic sequence, the common difference is the difference between any two consecutive terms.
The common difference can be positive, negative, or zero, depending on the pattern of the sequence.
Knowing the common difference is essential for finding the explicit formula and the recursive formula for an arithmetic sequence.
The common difference is a key characteristic that distinguishes arithmetic sequences from other types of sequences, such as geometric sequences.
Review Questions
Explain how the common difference is used to generate an arithmetic sequence.
In an arithmetic sequence, the common difference is the constant value that is added to each term to get the next term. For example, in the sequence 2, 5, 8, 11, 14, the common difference is 3, as each term is 3 more than the previous term. By knowing the first term and the common difference, you can use the explicit formula to find any term in the sequence.
Describe the relationship between the common difference and the explicit formula for an arithmetic sequence.
The common difference is a crucial component of the explicit formula for an arithmetic sequence. The explicit formula is given by $a_n = a_1 + (n-1)d$, where $a_n$ is the $n$th term, $a_1$ is the first term, and $d$ is the common difference. This formula allows you to find any term in the sequence by using the first term and the common difference, which is the constant value by which each term differs from the previous term.
Analyze how the sign of the common difference affects the behavior of an arithmetic sequence.
The sign of the common difference determines whether the sequence is increasing or decreasing. If the common difference is positive, the sequence is increasing, meaning each term is greater than the previous term. If the common difference is negative, the sequence is decreasing, meaning each term is less than the previous term. If the common difference is zero, the sequence is constant, meaning all terms are equal. Understanding the sign of the common difference is crucial for interpreting the behavior of an arithmetic sequence.
An arithmetic sequence is a sequence where the difference between consecutive terms is constant, known as the common difference.
Explicit Formula: The explicit formula for an arithmetic sequence allows you to find any term in the sequence using the first term and the common difference.
Recursive Formula: The recursive formula for an arithmetic sequence describes how to find each term in the sequence using the previous term and the common difference.