Problem 20 - Algebra and Trigonometry

Question

\(13-20\) . Determine whether the sequence is arithmetic. If 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 because the differences between terms are not constant.

Step 1: Identify the sequence

The sequence given is \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots \). We need to check if this sequence is arithmetic.

Step 2: Define an arithmetic sequence

An arithmetic sequence is defined as a sequence of numbers in which the difference between any two successive terms is constant and called the common difference. Mathematically, this means \( a_{n} - a_{n-1} = d \) for all terms \( a_n \).

Step 3: Calculate the differences

Calculate \( \frac{1}{2} - \frac{1}{3} = \frac{3-2}{6} = \frac{1}{6} \).Then calculate \( \frac{1}{3} - \frac{1}{4} = \frac{4-3}{12} = \frac{1}{12} \).Observe that these differences are not equal.

Step 4: Analyze the differences

Since \( \frac{1}{6} eq \frac{1}{12} \), the difference changes between the terms. Therefore, the sequence is not arithmetic.

Key Concepts

Common Difference

The common difference is a crucial part of understanding arithmetic sequences. It is the constant difference between each pair of successive terms in the sequence. When a sequence is arithmetic, this difference remains the same throughout. To check if a sequence is arithmetic, you subtract a term from the following one and compare the results with other pairs of terms.

For example, in an arithmetic sequence, if the first term is 2 and the second term is 4, then the common difference is found by computing 4 minus 2, which equals 2. As you move through the sequence, every term should maintain this difference, creating a uniform pattern. In the example provided, you noticed that the differences were not the same, illustrating that the sequence is not arithmetic.

Practicing to find the common difference is a foundational skill in mastering arithmetic sequences.

Sequence

A sequence refers to an ordered set of numbers written in a particular pattern. There can be various types of sequences, such as arithmetic, geometric, or even more complex ones. Sequences are a fundamental concept in mathematics, helping to understand patterns and relationships between numbers.

In the context of arithmetic sequences, knowing whether or not a sequence is arithmetic requires checking the differences between successive terms, as outlined in the given exercise. Arithmetic sequences have a specific pattern: they add or subtract the same value repeatedly to get from one term to the next, which is the common difference.

Being comfortable with identifying sequences and their types is a valuable skill that can be applied to other mathematical problems, including solving equations and modeling real-world situations.

Successive Terms

Successive terms in a sequence are simply terms that come one after another. In an arithmetic sequence, the relationship between successive terms is defined by the common difference. By this, any term in the sequence follows the previous one by consistently adding or subtracting the common difference.

In the analysis of the given sequence, you calculated the differences between successive terms like \( \frac{1}{2} \) and \( \frac{1}{3} \), which were not equal when checked with other successive terms such as \( \frac{1}{3} \) and \( \frac{1}{4} \).

Understanding successive terms and their differences is essential in verifying the type of sequence and in properly categorizing them. Such skills are particularly useful when dealing with problems that require extending a pattern or predicting beyond the given terms.