Nth Term

Definition

The nth term, also known as the general term, refers to the term in a sequence that corresponds to the nth position or index. It is a formula or expression that allows you to calculate any specific term in the sequence based on its position or index number.

5 Must Know Facts For Your Next Test

1. The nth term formula allows you to calculate any specific term in a sequence based on its position or index number.
2. In an arithmetic sequence, the nth term formula is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference.
3. In a geometric sequence, the nth term formula is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
4. The nth term formula is essential for understanding and working with sequences, as it enables you to predict and generate any term in the sequence.
5. Identifying the appropriate nth term formula is crucial for solving problems involving sequences, such as finding a specific term, the sum of the first $n$ terms, or the general pattern of the sequence.

Review Questions

• Explain how the nth term formula is used to generate terms in an arithmetic sequence.
  • In an arithmetic sequence, the nth term formula 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 substituting the position or index number (n) into the formula. 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 nth term formula is essential for understanding the pattern and generating any term in an arithmetic sequence.
• Describe how the nth term formula is used to generate terms in a geometric sequence.
  • In a geometric sequence, the nth term formula is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio. This formula allows you to calculate any specific term in the sequence by substituting the position or index number (n) into the formula. For example, if the first term is 2 and the common ratio is 3, the 4th term would be $a_4 = 2 \cdot 3^{4-1} = 54$. The nth term formula is essential for understanding the pattern and generating any term in a geometric sequence.
• Analyze how the understanding of the nth term formula can be applied to solve problems involving sequences.
  • The understanding of the nth term formula is crucial for solving a variety of problems related to sequences. By knowing the appropriate formula for the type of sequence (arithmetic or geometric), you can use the nth term to calculate specific terms, find the sum of the first n terms, or identify the general pattern of the sequence. For example, if you are given the first few terms of a sequence and asked to find the 20th term, you can use the nth term formula to determine the expression that generates the terms and then plug in the value of n = 20 to find the desired term. Similarly, the nth term formula can be used to find the sum of the first n terms of a sequence or to generalize the pattern of the sequence beyond the given information.

Related terms

Sequence: A sequence is an ordered list of elements, where each element is called a term, and the position of each term is indicated by an index number.

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.