Nth Term Formula

Definition

The nth term formula is a mathematical expression used to represent the general term or pattern in an arithmetic sequence. It allows for the prediction of any term in the sequence based on its position or index within the sequence.

5 Must Know Facts For Your Next Test

1. The nth term formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_n$ is the nth term, $a_1$ is the first term, and $d$ is the common difference.
2. The nth term formula allows you to find any term in the sequence by substituting the term's position (n) into the formula.
3. The common difference, $d$, is the constant value added to each term to get the next term in the sequence.
4. The first term, $a_1$, is the starting value of the sequence, and all other terms are derived from it using the common difference.
5. The nth term formula is a powerful tool for analyzing and predicting the behavior of arithmetic sequences.

Review Questions

• Explain how the nth term formula is used to generate terms in an arithmetic sequence.
  • The nth term formula, $a_n = a_1 + (n-1)d$, is used to generate any term in an arithmetic sequence. By substituting the desired position (n) into the formula, you can calculate the value of the nth term. For example, if the first term is 3 and the common difference is 2, the 5th term would be $a_5 = 3 + (5-1)2 = 11$. The formula allows you to easily determine any term in the sequence based on its position.
• Describe the relationship between the first term ($a_1$) and the common difference (d) in the nth term formula.
  • The first term, $a_1$, and the common difference, $d$, are both essential components of the nth term formula, $a_n = a_1 + (n-1)d$. The first term represents the starting value of the sequence, while the common difference determines the constant rate of change between consecutive terms. Together, these two values define the overall pattern of the arithmetic sequence, and the nth term formula allows you to calculate any term by combining these two key pieces of information.
• Analyze how the nth term formula can be used to determine the behavior and properties of an arithmetic sequence.
  • The nth term formula, $a_n = a_1 + (n-1)d$, provides a comprehensive understanding of the properties and behavior of an arithmetic sequence. By examining the formula, you can determine the first term ($a_1$) and the common difference ($d$), which are the fundamental characteristics that define the sequence. Additionally, the formula allows you to predict the value of any term ($a_n$) based on its position (n) within the sequence, enabling you to analyze the overall pattern and trends. This deep understanding of the sequence's structure and progression, as encapsulated in the nth term formula, is crucial for solving problems and making inferences about arithmetic sequences.

Related terms

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

Common Difference: The common difference is the constant value that is added to each term in an arithmetic sequence to obtain the next term.

Explicit Formula: The explicit formula, also known as the closed-form formula, is a formula that expresses each term of a sequence directly in terms of its position or index within the sequence.