key term - Constant Difference

Definition

A constant difference is a characteristic of an arithmetic sequence where the difference between consecutive terms in the sequence is a fixed, unchanging value. This constant difference is the common difference that defines the pattern of the sequence and allows for the prediction of future terms.

5 Must Know Facts For Your Next Test

1. The constant difference in an arithmetic sequence is the foundation for being able to predict future terms in the sequence.
2. Knowing the constant difference allows you to write the explicit formula for an arithmetic sequence, which 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.
3. The constant difference can be positive, negative, or zero, leading to sequences that are increasing, decreasing, or constant, respectively.
4. Arithmetic sequences with a constant difference of 0 are called constant sequences, as each term is the same value.
5. The constant difference is a key characteristic that distinguishes arithmetic sequences from other types of sequences, such as geometric sequences.

Review Questions

• Explain how the constant difference in an arithmetic sequence allows you to predict future terms.
  • The constant difference in an arithmetic sequence is the key to being able to predict future terms in the sequence. Since the difference between any two consecutive terms is a fixed value, you can use this constant difference to calculate the next term by adding the difference to the previous term. This pattern continues, allowing you to determine any term in the sequence once you know the first term and the common difference.
• Describe the relationship between the constant difference and the explicit formula for an arithmetic sequence.
  • The constant difference is directly linked to the explicit formula for an arithmetic sequence, which 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 constant difference). The constant difference, $d$, is used in the formula to calculate any term in the sequence based on its position relative to the first term. This formula allows you to predict future terms by using the constant difference.
• Analyze how the sign of the constant difference affects the behavior of an arithmetic sequence.
  • The sign of the constant difference in an arithmetic sequence determines whether the sequence is increasing, decreasing, or constant. If the constant difference is positive, the sequence is increasing, with each term being larger than the previous one. If the constant difference is negative, the sequence is decreasing, with each term being smaller than the previous one. If the constant difference is zero, the sequence is constant, with each term being the same value. The sign of the constant difference is a crucial factor in understanding the overall behavior and pattern of an arithmetic sequence.

Related terms

Arithmetic Sequence:

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is a constant value.

Common Difference:

The common difference is the constant value that is added to each term in an arithmetic sequence to generate the next term.

Explicit Formula:

The explicit formula for an arithmetic sequence allows you to calculate any term in the sequence based on the first term and the common difference.