Common difference

Definition

The common difference is a constant value that represents the fixed amount by which each term of an arithmetic sequence differs from the previous term. It plays a crucial role in defining the structure of arithmetic sequences, allowing for the identification of patterns and facilitating calculations related to their properties.

5 Must Know Facts For Your Next Test

1. The common difference can be positive, negative, or zero, affecting the direction and behavior of the arithmetic sequence.
2. It is typically denoted by the symbol 'd' and can be calculated by subtracting any term from its preceding term.
3. The formula for the n-th term of an arithmetic sequence is given by $$a_n = a_1 + (n-1)d$$, where $$a_1$$ is the first term and $$d$$ is the common difference.
4. In a graphical representation, an arithmetic sequence will appear as points evenly spaced along a straight line, where the spacing corresponds to the common difference.
5. Understanding the common difference is essential for solving problems related to series summation and analyzing patterns within sequences.

Review Questions

• How does the common difference define the structure of an arithmetic sequence?
  • The common difference is fundamental to an arithmetic sequence as it determines how each term relates to its predecessor. Each subsequent term is formed by adding the common difference to the previous term, creating a consistent pattern throughout the sequence. This relationship allows for easy calculation and understanding of the sequence's behavior, making it easier to predict future terms.
• Discuss how to calculate the n-th term of an arithmetic sequence using the common difference.
  • To calculate the n-th term of an arithmetic sequence, you use the formula $$a_n = a_1 + (n-1)d$$. Here, $$a_1$$ is the first term, $$d$$ represents the common difference, and $$n$$ is the position of the term you want to find. By substituting these values into the formula, you can quickly determine any specific term within the sequence, highlighting how integral the common difference is in this process.
• Evaluate how changing the common difference affects an arithmetic sequence and provide examples.
  • Changing the common difference alters the overall nature of an arithmetic sequence significantly. For example, if you have a sequence with a first term of 2 and a positive common difference of 3 (2, 5, 8, ...), it creates an increasing pattern. However, if you change the common difference to -2 (2, 0, -2, ...), it results in a decreasing pattern. Additionally, if you set the common difference to zero (2, 2, 2, ...), all terms remain constant. Thus, variations in the common difference can lead to diverse behaviors within sequences.

Related terms

arithmetic sequence: A sequence of numbers in which the difference between consecutive terms is constant, defined by the common difference.

first term: The initial term in an arithmetic sequence, often denoted as 'a' or 'a_1'. It serves as the starting point for generating subsequent terms.

n-th term: The term at position 'n' in a sequence, which can be calculated using the first term and the common difference.