A_n

Definition

The term $a_n$ represents the $n^{th}$ term in an arithmetic sequence. An arithmetic sequence is a sequence where the difference between any two consecutive terms is constant.

5 Must Know Facts For Your Next Test

1. The $n^{th}$ term of an arithmetic sequence can be calculated using the explicit formula: $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference.
2. The value of $a_n$ increases by the common difference $d$ with each successive term in the sequence.
3. The common difference $d$ can be found by subtracting any term from the term that immediately follows it.
4. Arithmetic sequences are useful for modeling real-world situations where quantities increase or decrease by a constant amount, such as savings accounts, loan payments, and population growth.
5. Analyzing the behavior of $a_n$ in an arithmetic sequence can provide insights into the overall pattern and allow for predictions about future terms.

Review Questions

• Explain how the term $a_n$ is used to represent the $n^{th}$ term in an arithmetic sequence.
  • The term $a_n$ represents the $n^{th}$ term in an arithmetic sequence. This means that $a_n$ is the value of the $n^{th}$ term, where $n$ is a positive integer. The value of $a_n$ can be calculated using the explicit formula $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference between consecutive terms. This formula allows you to determine the value of any term in the sequence without needing to know the previous terms.
• Describe how the common difference $d$ is related to the term $a_n$ in an arithmetic sequence.
  • The common difference $d$ is a crucial component in understanding the term $a_n$ within an arithmetic sequence. The common difference is the constant amount by which each term in the sequence differs from the previous term. This means that the value of $a_n$ increases or decreases by $d$ with each successive term. The common difference can be found by subtracting any term from the term that immediately follows it, and this value of $d$ can then be used in the explicit formula $a_n = a_1 + (n-1)d$ to calculate the value of the $n^{th}$ term $a_n$.
• Analyze how the behavior of the term $a_n$ can provide insights into the overall pattern of an arithmetic sequence.
  • Examining the behavior of the term $a_n$ in an arithmetic sequence can offer valuable insights into the underlying pattern of the sequence. Since the value of $a_n$ increases or decreases by the constant common difference $d$ with each successive term, observing the changes in $a_n$ can reveal the overall trend and allow for predictions about future terms. Additionally, the explicit formula $a_n = a_1 + (n-1)d$ demonstrates how the $n^{th}$ term is directly related to the first term $a_1$ and the common difference $d$, providing a deeper understanding of the sequence's structure and behavior.

Related terms

Arithmetic Sequence: A sequence where the difference between any two consecutive terms is constant.

Common Difference: The constant difference between any two consecutive terms in an arithmetic sequence.

Explicit Formula: A formula that allows you to calculate the $n^{th}$ term of an arithmetic sequence directly, without needing to know the previous terms.