$a_n = f(n)$

Definition

$a_n = f(n)$ is a mathematical expression that represents a sequence, where $a_n$ is the $n$-th term of the sequence and $f(n)$ is a function that defines the relationship between the term number $n$ and the value of the term $a_n$. This expression is fundamental in the study of sequences and their properties.

5 Must Know Facts For Your Next Test

1. The expression $a_n = f(n)$ is used to define an explicit sequence, where the $n$-th term of the sequence is directly calculated using the function $f(n)$.
2. The function $f(n)$ can be any mathematical expression or formula that takes the term number $n$ as input and produces the corresponding term value $a_n$.
3. Sequences defined by $a_n = f(n)$ are often called explicit or closed-form sequences, as the value of each term can be directly computed without the need for previous terms.
4. Explicit sequences can be linear, quadratic, exponential, or any other type of function, depending on the form of $f(n)$.
5. The expression $a_n = f(n)$ is a powerful tool in the study of sequences, as it allows for the analysis of sequence properties, such as convergence, divergence, and behavior.

Review Questions

• Explain the relationship between the term number $n$ and the term value $a_n$ in the expression $a_n = f(n)$.
  • In the expression $a_n = f(n)$, the term number $n$ is the independent variable that serves as the input to the function $f(n)$. The function $f(n)$ then produces the corresponding term value $a_n$, which is the $n$-th term of the sequence. This direct relationship between the term number and the term value is what defines an explicit sequence, allowing the value of each term to be calculated independently without the need for previous terms.
• Describe the different types of functions that can be used in the expression $a_n = f(n)$ to define various sequences.
  • The function $f(n)$ in the expression $a_n = f(n)$ can take many different forms, allowing for the creation of a wide variety of sequences. Some common types of functions used include linear functions ($f(n) = an + b$), quadratic functions ($f(n) = an^2 + bn + c$), exponential functions ($f(n) = a^n$), and even more complex mathematical expressions. The choice of function $f(n)$ determines the behavior and properties of the resulting sequence, such as whether it is arithmetic, geometric, or exhibits more complex patterns.
• Analyze the advantages of using the expression $a_n = f(n)$ to define sequences compared to other sequence representation methods.
  • The expression $a_n = f(n)$ offers several advantages in the study of sequences. First, it provides a concise and efficient way to represent a sequence, as the value of each term can be directly calculated using the function $f(n)$, without the need to list out all the terms individually. This makes it easier to analyze sequence properties, such as convergence, divergence, and behavior. Additionally, the explicit nature of $a_n = f(n)$ allows for the application of various mathematical techniques and theorems to study the sequence, which is not always possible with other sequence representation methods, such as recursive definitions. Overall, the $a_n = f(n)$ expression is a powerful tool in the field of sequence analysis and its applications in mathematics and related disciplines.

Related terms

Sequence: A sequence is an ordered list of elements, where each element is assigned a unique position or index, typically represented by a natural number.

Function: A function is a mathematical relationship that assigns a unique output value to each input value, following a specific rule or formula.

Recursive Sequence: A recursive sequence is a sequence where each term is defined in terms of the previous terms, often using a formula or an algorithm.