Problem 16 - Algebra and Trigonometry

Question

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots$$

Step-by-Step Solution

Answer

The sequence is not arithmetic.

Step 1: Identify the Sequence Pattern

The given sequence is \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots \). We need to determine the relationship between the terms to find out if it's arithmetic.

Step 2: Understand an Arithmetic Sequence

An arithmetic sequence is defined by a common difference \( d \) between consecutive terms, meaning that each term after the first is the sum of the previous term and \( d \). We express this as \( a_n = a_{n-1} + d \).

Step 3: Calculate Consecutive Term Differences

Calculate the differences between consecutive terms of the sequence. The first three differences are: \( \frac{1}{3} - \frac{1}{2} = -\frac{1}{6} \), \( \frac{1}{4} - \frac{1}{3} = -\frac{1}{12} \), and \( \frac{1}{5} - \frac{1}{4} = -\frac{1}{20} \).

Step 4: Check for Consistent Differences

Notice that the difference between consecutive terms isn't constant: \(-\frac{1}{6} \), \(-\frac{1}{12} \), \(-\frac{1}{20} \). Since the differences are not the same, the sequence is not arithmetic.

Key Concepts

Common Difference

When we talk about arithmetic sequences, a common term you will encounter is "common difference." This concept refers to the consistent difference between consecutive terms in a sequence. For a sequence to be considered arithmetic, this difference must remain the same throughout all the terms.
To visualize this, think about a simple sequence such as 2, 4, 6, 8, ... Here, the common difference is 2, because each term is 2 more than the one before it.

• Example: In the sequence 3, 6, 9, 12, the common difference is 3.
• Verification: You can verify a sequence is arithmetic by calculating the difference of consecutive terms and checking if they stay constant.

Identifying and understanding the common difference is crucial for figuring out whether a sequence is arithmetic.

Consecutive Terms

In any sequence, consecutive terms are simply terms that follow one after the other. Understanding the role of consecutive terms in sequences is essential to figure out patterns. In arithmetic sequences, examining consecutive terms helps us identify the common difference.
Suppose we have a series 5, 10, 15, 20. Each number is a consecutive term that follows the previous one in an orderly fashion. Observing these consecutive terms allows you to spot the pattern: each term is 5 more than the last one.

• Important Note: In arithmetic sequences, the formula for finding any term is based on its consecutive position: \[ a_n = a_1 + (n-1) imes d \] This involves the first term \( a_1 \) and the common difference \( d \).
• Practice: To make this clearer, try using consecutive terms in different sequences to calculate the common difference on your own.

By focusing on consecutive terms, you can more effectively determine if a sequence is arithmetic.

Sequence Patterns

Sequence patterns help us identify the rules that define the sequence. In the context of arithmetic sequences, the pattern is the addition of a consistent number, known as the common difference, to each term to get to the next. Understanding these patterns can make solving these types of problems easier and more intuitive.
In our example, the sequence \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} \), the pattern is not arithmetic because the differences between terms \( -\frac{1}{6}, -\frac{1}{12}, -\frac{1}{20} \) are not constant. Recognizing sequence patterns involves:

• Checking the differences between consecutive terms.
• Verifying if changes in terms happen by a consistent increment or decrement.
• Understanding that not all sequences will fit the arithmetic model, highlighting the importance of observing the pattern closely.

Grasping sequence patterns is key to mastering sequence-based problems, whether they confirm to the arithmetic type or not.