Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the sequence is arithmetic. If so find the common difference. $$ \\{11.4,9.3,7.2,5.1,3, \ldots\\} $$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of -2.1.

Step 1: Understand the Definition of an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is the same. This constant difference is called the common difference.

Step 2: Calculate the Differences Between Consecutive Terms

To determine if the sequence is arithmetic, we calculate the differences between consecutive terms: - Difference between second and first term: 9.3 - 11.4 = -2.1 - Difference between third and second term: 7.2 - 9.3 = -2.1 - Difference between fourth and third term: 5.1 - 7.2 = -2.1 - Difference between fifth and fourth term: 3 - 5.1 = -2.1

Step 3: Verify Consistency of the Differences

Since all calculated differences (-2.1) are the same, the sequence is arithmetic. This consistent difference indicates that the sequence has a common difference.

Step 4: Identify the Common Difference

The common difference for the given arithmetic sequence is -2.1.

Key Concepts

Understanding the Common Difference in Arithmetic Sequences

A key feature of arithmetic sequences is the **common difference**. This is the fixed amount that you add (or subtract) to one term to get to the next term in the series. To find the common difference, you simply subtract any term in the sequence from the term that follows it.

If every result from these subtractions is the same, it confirms the sequence is arithmetic:

• In our given sequence \( \{11.4, 9.3, 7.2, 5.1, 3, \ldots\} \), the common difference \( d \) is computed as follows:
• \( 9.3 - 11.4 = -2.1 \)
• \( 7.2 - 9.3 = -2.1 \)
• \( 5.1 - 7.2 = -2.1 \)
• \( 3 - 5.1 = -2.1 \)

The consistency of a common difference, in this case, \(-2.1\), affirms the sequence's arithmetic nature. Understanding and identifying the common difference allows us to predict future terms in the series and is an essential tool in sequence-related problems.

Defining an Arithmetic Sequence

An **arithmetic sequence** is a series made up of numbers where the difference between any two consecutive terms is constant. This type of sequence is quite common in various math problems because it follows a simple and predictable pattern.

The sequence can be expressed using the formula:

• For any sequence term \( a_n \), the equation is \( a_n = a_1 + (n-1)d \).
• \( a_1 \) represents the first term of the sequence.
• \( d \) is the common difference.

By understanding this formula, you can determine any term in the sequence without listing all preceding terms. This becomes particularly useful when dealing with large sequences or finding terms that are far apart.

Exploring Consecutive Terms in Arithmetic Sequences

When working with an arithmetic sequence, **consecutive terms** are the terms that come one after the other. The difference between these terms is what we call the common difference. In arithmetic sequences, this gap is always fixed, which gives the sequence its regularity.

In our sequence \( \{11.4, 9.3, 7.2, 5.1, 3, \ldots\} \):

• The consecutive terms are 11.4 and 9.3, 9.3 and 7.2, 7.2 and 5.1, and so on.
• The difference between each pair is \(-2.1\), confirming they are equally spaced.

Understanding the notion of consecutive terms helps in visualizing how each part of the sequence is constructed. It ensures you know how to move from one term to the next, keeping the sequence's structure intact.