Problem 19 - Algebra and Trigonometry

Question

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: Understanding the Arithmetic Sequence

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. This difference is called the common difference.

Step 2: Identify Consecutive Terms

Look at the sequence given: \(2.6, 4.3, 6.0, 7.7, \ldots\). The consecutive terms are pairs like \((2.6, 4.3)\), \((4.3, 6.0)\), and \((6.0, 7.7)\).

Step 3: Find the Difference Between Consecutive Terms

Subtract the first term from the second term: \(4.3 - 2.6 = 1.7\). Check the next pair: \(6.0 - 4.3 = 1.7\). Lastly, check the pair \(7.7 - 6.0 = 1.7\).

Step 4: Check for Consistent Difference

Since all differences are \(1.7\), the sequence has a consistent common difference of \(1.7\). This confirms the sequence is arithmetic.

Key Concepts

Common Difference

In an arithmetic sequence, the term 'common difference' refers to the constant value obtained when subtracting one term from the next. It is a fundamental component of the sequence, dictating the pattern followed by the terms. Consider the sequence given: \(2.6, 4.3, 6.0, 7.7, \ldots \) . Here, to determine if it's arithmetic, we calculate the difference between the consecutive pairs. For example:

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

The equality of these differences, which is 1.7, confirms the whole sequence follows a constant pattern. This constant is the common difference that makes the sequence arithmetic. Identifying and verifying the common difference is crucial when working with such sequences.

Consecutive Terms

Consecutive terms in a sequence are pairs of terms that follow one after another. In an arithmetic sequence, the difference between each pair of consecutive terms must remain constant for the sequence to be classified as such. Let's break it down using the sequence: \(2.6, 4.3, 6.0, 7.7, \ldots \). Here:

• 2.6 and 4.3 are consecutive.
• 4.3 and 6.0 are consecutive.
• 6.0 and 7.7 are consecutive.

The consistent calculation of the common difference across each consecutive term confirms the arithmetic nature of the sequence. By using consecutive terms to compare differences, you can easily establish whether a sequence is arithmetic. Simply ensure that each calculation between every pair yields the same result.

Sequence Analysis

Sequence analysis involves examining the structure and pattern of a series of numbers to identify specific characteristics, such as whether a sequence is arithmetic. This process begins by listing out the terms, which helps in pinpointing any mathematical relation among the consecutive terms. In our sequence: \(2.6, 4.3, 6.0, 7.7, \ldots \), we notice each number is progressively increasing at a steady rate. The next step in sequence analysis is computing the difference between each consecutive pair of terms to verify if it remains consistent. In our example, the common difference of 1.7 across all pairs serves as a key indicator that the sequence is arithmetic. Analyzing sequences not only helps in understanding the present terms but also assists in predicting future numbers. By continuing the sequence using the common difference, we can foresee upcoming terms, strengthening one's ability to work with numeric relationships.