key term - Nth Term

Definition

The nth term, also known as the general term, refers to the expression that defines the value of a term in a sequence at any given position or index. It allows for the calculation of any 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 the value of any 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 can be used to find the value of any term in the sequence, as well as to determine the total number of terms in the sequence.
5. Understanding the nth term formula is crucial for solving problems involving sequences, such as finding the sum of the first $n$ terms or determining the position of a specific term in the sequence.

Review Questions

• Explain how the nth term formula is used to calculate the value of a specific term in 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. To find the value of the nth term, you simply plug in the value of $n$ (the position or index number of the term you want to find) and the values of $a_1$ and $d$ into the formula. This allows you to calculate the value of any term in the sequence based on its position.
• 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 is the way they calculate the value of a term based on its position in the sequence. For an arithmetic sequence, the formula is $a_n = a_1 + (n-1)d$, where $d$ is the common difference between terms. For a geometric sequence, the formula is $a_n = a_1 \cdot r^{n-1}$, where $r$ is the common ratio between terms. In an arithmetic sequence, the value of each term increases or decreases by a constant amount, while in a geometric sequence, the value of each term is multiplied by a constant ratio.
• Explain how the nth term formula can be used to determine the total number of terms in a sequence.
  • The nth term formula can be rearranged to solve for the index number $n$, which represents the total number of terms in the sequence. For example, in an arithmetic sequence with the formula $a_n = a_1 + (n-1)d$, if you know the value of the last term $a_n$ and the common difference $d$, you can solve for $n$ to find the total number of terms. Similarly, in a geometric sequence with the formula $a_n = a_1 \cdot r^{n-1}$, if you know the value of the last term $a_n$ and the common ratio $r$, you can solve for $n$ to determine the total number of terms in the sequence.

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.