Problem 1 - College Algebra with Corequisite Support

Question

What is an arithmetic sequence?

Step-by-Step Solution

Answer

An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant.

Step 1: Introduction to Sequences

A sequence is an ordered list of numbers following a specific pattern or rule. Each number in the list is called a term.

Step 2: Understanding Arithmetic Sequences

An arithmetic sequence is a specific type of sequence where the difference between consecutive terms is constant. This constant difference is called the 'common difference' and is usually denoted by the letter \(d\).

Step 3: Identifying the General Form

The general form of an arithmetic sequence is given by the equation \(a_n = a_1 + (n-1)d\), where \(a_n\) is the \(n\)-th term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

Key Concepts

Sequences

A sequence is simply an ordered list of numbers. These numbers are called terms and they follow a specific rule or pattern. Sequences can be found everywhere, from the arrangement of numbers in mathematics to patterns observed in nature, like the spirals of a sunflower. Understanding sequences is crucial as they form the basis for more advanced concepts in mathematics.

There are various types of sequences, such as arithmetic, geometric, and harmonic. Each type follows a unique rule that defines the relationship between its terms. Sequences help us analyze patterns, predict outcomes, and solve problems in a structured manner. By recognizing the pattern, we can determine unknown terms or extend the sequence indefinitely.

Common Difference

In arithmetic sequences, the common difference is a key concept. It can be seen as the 'glue' that holds the sequence together, determining the spacing between terms. The common difference, denoted by the letter \(d\), refers to the constant amount added or subtracted to get from one term to the next in an arithmetic sequence.

For example, in the sequence 2, 5, 8, 11, the common difference \(d\) is 3. This is because each term is 3 units more than the previous term. Knowing the common difference allows you to easily create additional terms in the sequence or verify if a list of numbers forms an arithmetic sequence by checking the consistency of this difference.

General Form of a Sequence

The general form of an arithmetic sequence is essential to calculate any term in the sequence without listing all preceding terms. This is a powerful tool as it allows you to find any term by just knowing the first term, the common difference, and the position of the term needed. The formula used is \(a_n = a_1 + (n-1)d\).

Here, \(a_n\) represents the nth term you wish to find, \(a_1\) is the first term in the sequence, \(n\) is the term number, and \(d\) is the common difference. This formula simplifies the process of finding terms and helps in solving problems related to arithmetic sequences efficiently. For example, in the sequence 3, 6, 9, 12, if you want to find the 10th term, you would use the formula: \(a_{10} = 3 + (10-1) \times 3 = 30\).