Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-8,\) determine if each sequence is an arithmetic sequence. If the sequence is arithmetic, find the common difference. $$ 20,15,10,5,0, \dots $$

Step-by-Step Solution

Answer

Yes, the sequence is arithmetic with a common difference of -5.

Step 1: Understand the Sequence

Identify the given sequence: \(20, 15, 10, 5, 0, \ldots\). Notice that the sequence decreases by 5 at each step.

Step 2: Find the Common Differences

Compute the differences between consecutive terms: \(15 - 20 = -5\), \(10 - 15 = -5\), \(5 - 10 = -5\), and \(0 - 5 = -5\).

Step 3: Determine if Sequence is Arithmetic

Since the difference between consecutive terms is constant (-5), the sequence is arithmetic by definition.

Step 4: Find the Common Difference

The common difference, \(d\), of the arithmetic sequence is \(-5\).

Key Concepts

Understanding the Common Difference

In arithmetic sequences, the most important concept is the common difference. This is a fixed number that demonstrates how much each term in the sequence increases or decreases from the previous one. Here, if we take the example of given sequence: \(20, 15, 10, 5, 0, \ldots\), and compute the differences between consecutive terms, we find that each difference is \(- 5\). This constant value, \(- 5\), is the common difference.

• It is denoted by \(d\).
• The common difference can be positive, negative, or zero.

So, when determining if a sequence is arithmetic, check if a consistent difference exists between terms. If it does, that number is the common difference.

Spotting Consecutive Terms

Consecutive terms in a sequence are simply terms that follow one after the other without skipping any intermediate terms. By examining this closely, you reaffirm the structure and pattern of the sequence. Using our sequence \(20, 15, 10, 5, 0, \ldots\):

• 20 and 15 are consecutive terms.
• Similarly, 15 and 10 are consecutive as well as 10 and 5, and so on.

Recognizing consecutive terms helps in verifying the consistency of the common difference in arithmetic sequences. Each step moves from one term to the next with the same change, grounding the sequence firmly in arithmetic progression.

Array of a Sequence

A sequence is an ordered list of numbers following a certain pattern. When terms follow a strong and identifiable regularity, they can define whether a sequence is arithmetic, geometric, or neither. Sequences are defined in distinctive ways:

• Arithmetic sequences add a fixed number to get from one term to the next.
• Geometric sequences multiply terms by a constant.

The sequence \(20, 15, 10, 5, 0, \ldots\) clearly requires each term to be reduced by \(5\). Thus, it highlights a defined pattern, making it an arithmetic sequence. Recognizing the logical construction of a sequence is pivotal in identifying its type.

Introduction to Arithmetic Progression

An arithmetic progression (AP) is a special kind of sequence where the gap between each pair of consecutive terms is the same. This consistent gap is what we call the common difference.

• The formula for the \(n\)-th term of an arithmetic progression is \(a_n = a_1 + (n-1)\cdot d\).

This mathematical definition allows us to generalize and predict any term within the sequence. Using the example of our sequence \(20, 15, 10, 5, 0, \ldots\): it starts with 20, and the common difference \(d\) is \(-5\), allowing us to pinpoint any future or past term without listing them all out.Being comfortable with the concept of arithmetic progression equips you with a powerful tool to analyze linear patterns in numeric series.