Sequence

Definition

A sequence is an ordered list of elements, typically numbers or mathematical objects, that follow a specific pattern or rule. Sequences are fundamental concepts in mathematics and have important applications in various fields, including arithmetic sequences and series.

5 Must Know Facts For Your Next Test

1. The terms in a sequence are typically denoted by $a_1, a_2, a_3, \dots, a_n$, where $a_n$ represents the $n$-th term in the sequence.
2. An arithmetic sequence is defined by a common difference, $d$, between consecutive terms, and can be expressed as $a_n = a_1 + (n-1)d$.
3. The sum of the first $n$ terms in an arithmetic sequence is given by the formula $S_n = \frac{n}{2}[2a_1 + (n-1)d]$.
4. A series is the sum of the terms in a sequence, and can be denoted using the summation notation $\sum_{i=1}^n a_i$, where $a_i$ is the $i$-th term in the sequence.
5. The notation $a_n$ is used to represent the $n$-th term in a sequence, while $\sum_{i=1}^n a_i$ represents the sum of the first $n$ terms in a series.

Review Questions

• Explain the relationship between a sequence and an arithmetic sequence, and how the common difference is used to define an arithmetic sequence.
  • A sequence is a general ordered list of elements, while an arithmetic sequence is a specific type of sequence where the difference between any two consecutive terms is constant. This constant difference is known as the common difference, $d$, and it can be used to define the $n$-th term of an arithmetic sequence as $a_n = a_1 + (n-1)d$, where $a_1$ is the first term in the sequence. The common difference allows you to easily generate the terms of an arithmetic sequence based on the first term and the constant difference between terms.
• Describe how the summation notation $\sum_{i=1}^n a_i$ is used to represent the sum of the terms in a sequence, and explain the relationship between a sequence and a series.
  • The summation notation $\sum_{i=1}^n a_i$ is used to represent the sum of the first $n$ terms in a sequence, where $a_i$ represents the $i$-th term. This sum is known as a series, which is the accumulation of the terms in a sequence. The relationship between a sequence and a series is that a sequence is an ordered list of elements, while a series is the sum of those elements. The series can be denoted using the summation notation, which provides a concise way to represent the sum of the terms in a sequence.
• Analyze how the notation used to represent sequences and series, such as $a_n$ and $\sum_{i=1}^n a_i$, helps to convey information about the structure and behavior of these mathematical concepts.
  • The notation used to represent sequences and series is designed to provide a clear and concise way to communicate the underlying structure and properties of these mathematical concepts. The notation $a_n$ is used to represent the $n$-th term in a sequence, which allows you to easily identify and manipulate individual terms within the sequence. The summation notation $\sum_{i=1}^n a_i$ represents the sum of the first $n$ terms in a sequence, which is the series. This notation helps to convey the accumulative nature of a series and the relationship between the individual terms in the sequence and the overall sum. The use of these standardized notations enables mathematicians and students to efficiently work with and understand sequences and series, and to apply these concepts in various mathematical and scientific contexts.

Related terms

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

Series: A series is the sum of the terms in a sequence, often represented by the symbol $\sum$.

Notation: Notation refers to the symbolic representation used to express mathematical concepts, such as sequences and series.