Nth Term Formula

Definition

The nth term formula, also known as the general term formula, is a mathematical expression that describes the pattern of a sequence and allows for the calculation of any specific term within that sequence. This formula is a crucial tool in understanding and working with two important types of sequences: arithmetic sequences and geometric sequences.

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 $a_1$ is the first term and $d$ is the common difference.
2. The nth term formula for a geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
3. The nth term formula allows you to calculate any specific term in a sequence without needing to generate the entire sequence.
4. Understanding the nth term formula is essential for solving problems involving the sum of a finite or infinite arithmetic or geometric series.
5. The nth term formula can be used to determine the behavior and properties of a sequence, such as whether it is increasing, decreasing, or constant.

Review Questions

• Explain how the nth term formula is used to describe the pattern of an arithmetic sequence.
  • The nth term formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference. This formula allows you to calculate any specific term in the sequence by knowing the first term and the common difference. The formula captures the pattern of the sequence, where each term is obtained by adding the common difference to the previous term. This formula is essential for understanding the behavior and properties of arithmetic sequences, such as whether the sequence is increasing, decreasing, or constant.
• Describe the differences between the nth term formulas for arithmetic and geometric sequences.
  • The key difference between the nth term formulas for arithmetic and geometric sequences lies in the way the terms are generated. For an arithmetic sequence, the nth term formula is $a_n = a_1 + (n-1)d$, where the common difference $d$ is added to the previous term to obtain the next term. In contrast, the nth term formula for a geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where the common ratio $r$ is multiplied by the previous term to obtain the next term. This fundamental difference in the way the terms are generated leads to distinct patterns and properties between arithmetic and geometric sequences, which are crucial to understand when working with these types of sequences.
• Analyze how the nth term formula can be used to determine the behavior and properties of a sequence.
  • The nth term formula provides a powerful tool for analyzing the behavior and properties of a sequence. By examining the structure of the formula, you can determine key characteristics of the sequence. For example, in an arithmetic sequence, if the common difference $d$ is positive, the sequence is increasing; if $d$ is negative, the sequence is decreasing; and if $d$ is zero, the sequence is constant. Similarly, in a geometric sequence, if the common ratio $r$ is greater than 1, the sequence is increasing; if $r$ is between 0 and 1, the sequence is decreasing; and if $r$ is 1, the sequence is constant. Understanding these relationships between the nth term formula and the sequence's behavior is crucial for solving problems and making inferences about the sequence.

Related terms

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

Geometric Sequence: A geometric sequence is a sequence where the ratio between any two consecutive terms is constant.

Common Difference: In an arithmetic sequence, the common difference is the constant value that is added to each term to get the next term.