Problem 7 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether or not the sequence is arithmetic. If it is, find the common difference. $$3, \frac{5}{2}, 2, \frac{3}{2}, 1, \ldots$$

Step-by-Step Solution

Answer

Yes, the sequence is arithmetic. The common difference is -1/2.

Step 1: Identify the difference between consecutive terms

The difference between the first term and the second term is \(3 - \frac{5}{2} = -\frac{1}{2}\). The difference between the second term and the third is \(\frac{5}{2} - 2 = -\frac{1}{2}\). The difference between the third term and the forth one is \(2 - \frac{3}{2} = -\frac{1}{2}\). The difference between the forth term and the fifth one is \(\frac{3}{2} - 1 = -\frac{1}{2}\).

Step 2: Check if the difference is consistent

The difference between each consecutive term is always -\frac{1}{2}, which is consistent.

Step 3: Confirm if the sequence is arithmetic

An arithmetic sequence is defined by a common difference between consecutive terms. Since the difference is consistent, the sequence is arithmetic.

Step 4: Find the common difference

The common difference of the arithmetic sequence is -\frac{1}{2}, which is the difference identified in Step 1.

Key Concepts

Common Difference

When dealing with arithmetic sequences, the term "common difference" is the key concept that helps identify the sequence type. An arithmetic sequence grows or declines by adding a fixed amount to each term. This fixed amount is called the common difference.

To determine if a sequence is arithmetic, look at the difference between consecutive terms. If this difference is the same each time, it is the common difference. In our sequence, starting from 3, the difference between terms, \(-\frac{1}{2}\),\ is always the same.

• Calculate the difference by subtracting a term from the previous term.
• If all differences are equal, you've found the common difference.

For example, if the difference between each pair of consecutive terms is -\(\frac{1}{2}\), like in the exercise given, the common difference is consistently -\(\frac{1}{2}\), confirming the arithmetic nature.

Sequence Patterns

Recognizing patterns in sequences can streamline the identification of arithmetic sequences. Pattern recognition involves identifying regular additions or subtractions that maintain uniformity across a sequence.

• In arithmetic sequences, look for a repetitive addition or subtraction.
• Compare each term to the next to see if the pattern holds.

In the provided sequence \(3, \frac{5}{2}, 2, \frac{3}{2}, 1, \ldots\), observe each succeeding term decreases by \(-\frac{1}{2}\). This pattern is uniform and shows a common arithmetic progression.

Spotting such patterns can simplify understanding and defining sequences, allowing quick checks for arithmetic properties.

Mathematics Problem-Solving

Mathematics problem-solving with sequences often revolves around systematic approaches. When working with sequences like the one in the exercise, a structured method makes it easier to analyze and solve.

Here's a simple process:

• Identify each term and calculate the differences consecutively in sequences.
• Check the consistency of calculated differences, ensuring they're identical.
• Confirm properties, such as identifying if it's arithmetic with a fixed pattern.

By checking these steps, solving mathematical problems about sequences becomes more accessible and organized. You ensure that every sequence is properly identified and understood by considering the problem within a framed approach, just as shown in the solution steps from the original exercise.

Effective problem-solving involves clear steps and logical reasoning, allowing for insightful solutions to seemingly complex problems.