Problem 15 - Algebra and Trigonometry

Question

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$2.6,4.3,6.0,7.7, \ldots$$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of 1.7.

Step 1: Observe the Sequence

Look at the given sequence: 2.6, 4.3, 6.0, 7.7, ... We need to determine if this sequence is arithmetic.

Step 2: Calculate the Differences

In an arithmetic sequence, the difference between consecutive terms is constant. Let's calculate the differences between the first few terms in the sequence.1. Difference between 4.3 and 2.6: \[4.3 - 2.6 = 1.7\]2. Difference between 6.0 and 4.3: \[6.0 - 4.3 = 1.7\]3. Difference between 7.7 and 6.0: \[7.7 - 6.0 = 1.7\]

Step 3: Determine if the Sequence is Arithmetic

Since all consecutive differences are the same, the sequence is arithmetic.

Step 4: Identify the Common Difference

The common difference in an arithmetic sequence is the consistent difference we calculated between terms. In this case, the common difference is 1.7.

Key Concepts

Understanding Common Difference

In an arithmetic sequence, one of the most important aspects to identify is the **common difference**. This term refers to the constant amount you add or subtract to get from one term to the next in the sequence.
For example, consider the sequence from the exercise: 2.6, 4.3, 6.0, 7.7, ...
To find the common difference, calculate the difference between each pair of consecutive terms:

• Between 4.3 and 2.6: \(4.3 - 2.6 = 1.7\)
• Between 6.0 and 4.3: \(6.0 - 4.3 = 1.7\)
• Between 7.7 and 6.0: \(7.7 - 6.0 = 1.7\)

Since these differences are constant, the common difference is 1.7. This uniformity is what confirms the sequence as arithmetic.
Recognizing the common difference is crucial because it helps us predict future terms and analyze the structure of the sequence more efficiently.

The Role of Consecutive Terms

**Consecutive terms** in an arithmetic sequence are successive elements where the difference between each term is the same. For instance, in our sequence, each term follows its predecessor by the common difference of 1.7.
Let's look at the sequence: 2.6, 4.3, 6.0, 7.7, ...
Here, 2.6 is followed by 4.3, which then is followed by 6.0, and so on.
The relationship between these terms can be described mathematically as:

\(a_{n+1} = a_n + d\)

where \(a_{n+1}\) is the next term, \(a_n\) is the current term, and \(d\) is the common difference (1.7 in our case).
The concept of consecutive terms helps us identify the pattern governing the sequence, which is fundamental in confirming if a sequence is truly arithmetic.

Sequence Analysis and Verification

When performing **sequence analysis**, it is vital to verify whether a sequence follows a specific pattern or rule—in this case, whether it is arithmetic. The logic here involves checking the differences between the terms.
With sequences, we achieve this analysis by examining the differences between consecutive terms to see if they remain consistent. For our sequence (2.6, 4.3, 6.0, 7.7, ...), we calculated each difference to be 1.7. Here’s the process again:

• Between 4.3 and 2.6: \(4.3 - 2.6 = 1.7\)
• Between 6.0 and 4.3: \(6.0 - 4.3 = 1.7\)
• Between 7.7 and 6.0: \(7.7 - 6.0 = 1.7\)

Because the differences are all identical, we conclude that the sequence is arithmetic.
Sequence analysis is a useful skill in mathematics because it allows us to systematically understand and predict behavior within a set of numbers. It ensures that we can recognize and extend the sequence by maintaining the identified pattern.