Question

Determine whether the sequence is arithmetic, geometric, or neither. \(-7,-1,5,11,17, \ldots\)

Short answer

The given sequence is an arithmetic sequence.

Step-by-step solution

Calculate the difference between consecutive terms

The first step involves finding the differences between consecutive terms: \(-1 - (-7) = 6\), \(5 - (-1) = 6\), \(11 - 5 = 6\), and \(17 - 11 = 6\). The difference sequence contains a constant term \(6\).

Create the ratio of consecutive terms.

Even though from Step 1 it seems clear that the sequence is arithmetic, for the sake of completeness, the ratio of consecutive terms is calculated because geometric sequences show a constant ratio. The ratios are: |-1 / -7| = 0.14, |5 / -1| = -5, |11 / 5| = 2.2, |17 / 11| = 1.55. Here, ratios aren't constant.

Identify the type of sequence

Since the difference is constant and the ratio is not constant, we have an arithmetic sequence. The sequence \( -7,-1,5,11,17, \ldots \) is an arithmetic sequence with a common difference of 6.

Key concepts

Sequence Analysis

When we analyze a sequence, we are essentially looking for patterns that can help us to determine the nature of the sequence. This can be an essential skill in mathematics and many scientific fields because sequences are used to model real-world situations, from population growth to computer algorithms.

Steps in Sequence Analysis

Sequence analysis in mathematics commonly involves:

• Identifying whether the sequence progresses with a pattern
• Determining the rule that generates the sequence
• Predicting future terms in the sequence
• Calculating specific terms when given the position in the sequence

This process equips students with the tools to handle various sequence-related problems efficiently, from simple pattern recognition to more complex mathematical modelling.

Arithmetic Progression

An arithmetic progression, also known as an arithmetic sequence, is one of the most common types of sequences in mathematics. It is characterized by a constant difference between consecutive terms.

Features of Arithmetic Progression

An arithmetic progression:

• Begins with an initial term
• Continues with additional terms added by a consistent value, known as the common difference
• Can be represented by the formula for the nth term: \( a_n = a_1 + (n - 1)d \), where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( d \) is the common difference

This straightforward pattern not only simplifies the process of finding terms in the sequence but also provides a clear structure for summarizing data and modelling linear trends.

Common Difference

The common difference in an arithmetic sequence is the key feature that defines its progression. It is the fixed amount that each term increases by from one to the next.

Importance of Common Difference

The common difference:

• Is the difference obtained when subtracting one term from the following term
• Determines the rate at which the sequence increases or decreases
• Is consistent throughout the whole sequence
• Can be positive, negative, or even zero, which would make the sequence constant

Understanding the common difference allows students to decipher the pattern of an arithmetic sequence quickly and is critical in solving related problems, including those involving sums of arithmetic series and real-world application problems.