The common difference is the constant value by which each term in an arithmetic sequence differs from the previous term. It represents the consistent rate of change between consecutive terms in the sequence.
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The common difference is denoted by the variable $d$ and represents the constant rate of change in an arithmetic sequence.
To determine the common difference, you can subtract any two consecutive terms in the sequence.
The common difference allows you to predict the value of any term in an arithmetic sequence, given the first term and the common difference.
Arithmetic sequences with a positive common difference are called increasing sequences, while those with a negative common difference are called decreasing sequences.
The common difference is a crucial concept in understanding the patterns and behaviors of arithmetic sequences.
Review Questions
Explain how the common difference relates to the definition of an arithmetic sequence.
The common difference is the defining characteristic of an arithmetic sequence. By definition, an arithmetic sequence is a sequence where the difference between any two consecutive terms is constant. This constant difference is the common difference, denoted by the variable $d$. The common difference represents the consistent rate of change between terms in the sequence, allowing you to predict the value of any term once you know the first term and the common difference.
Describe how the sign of the common difference affects the behavior of an arithmetic sequence.
The sign of the common difference determines whether an arithmetic sequence is increasing or decreasing. If the common difference $d$ is positive, the sequence is increasing, meaning each term is greater than the previous term. If the common difference $d$ is negative, the sequence is decreasing, meaning each term is less than the previous term. The common difference, along with the first term, allows you to predict the value of any term in the sequence, making it a crucial concept in understanding the patterns and behaviors of arithmetic sequences.
Analyze how the common difference can be used to generate any term in an arithmetic sequence, given the first term.
$$The common difference, $d$, is the key to generating any term in an arithmetic sequence, given the first term. The formula for the $n^{th}$ term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference. By plugging in the known values of $a_1$ and $d$, you can calculate the value of any term $a_n$ in the sequence. This allows you to predict the behavior and patterns of the sequence, making the common difference a fundamental concept in understanding arithmetic sequences.$$
A sequence is an ordered list of elements, where each element is called a term.
Constant Rate of Change: A constant rate of change is a fixed value by which a quantity increases or decreases between consecutive terms in a sequence.