Problem 13 - Algebra and Trigonometry

Question

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ 5,8,11,14, \dots $$

Step-by-Step Solution

Answer

Yes, it is arithmetic. Common difference: 3.

Step 1: Understanding the Problem

We are given a sequence of numbers: 5, 8, 11, 14, ... We need to determine if this sequence is arithmetic and, if so, find its common difference.

Step 2: Defining an Arithmetic Sequence

An arithmetic sequence is characterized by a constant difference between consecutive terms, called the common difference. We need to find the difference between consecutive terms in our sequence.

Step 3: Finding the Difference for First Pair of Terms

Calculate the difference between the second term (8) and the first term (5): \[ 8 - 5 = 3 \].

Step 4: Finding the Difference for Second Pair of Terms

Calculate the difference between the third term (11) and the second term (8): \[ 11 - 8 = 3 \].

Step 5: Finding the Difference for Third Pair of Terms

Calculate the difference between the fourth term (14) and the third term (11): \[ 14 - 11 = 3 \].

Step 6: Conclusion on Sequence Type

Since the difference between consecutive terms is constant (3), the sequence is arithmetic.

Step 7: Identifying the Common Difference

The common difference of the arithmetic sequence is 3.

Key Concepts

Common Difference in Arithmetic Sequences

An arithmetic sequence is one where each term after the first is found by adding a constant, known as the "common difference," to the previous term. To determine if a sequence is arithmetic, you must verify that the difference between any two consecutive terms is consistent throughout the sequence.
For example, in the sequence 5, 8, 11, 14, ..., you calculate:

• Second term (8) - First term (5) = 3
• Third term (11) - Second term (8) = 3
• Fourth term (14) - Third term (11) = 3

As each difference equals 3, it confirms the sequence is arithmetic with a common difference of 3. Whenever you find a consistent difference, it's like unraveling a secret pattern that guides the sequence. This pattern provides predictability, allowing for easy calculations of future terms.

Sequence Analysis

Sequence analysis involves evaluating sequences to understand their underlying patterns and properties. By examining the differences in an arithmetic sequence, you can quickly determine its type. For arithmetic sequences, focusing on the common difference is crucial.
When analyzing, look for:

• Consistency: Check that the difference between consecutive terms remains constant.
• Pattern Recognition: Identifying repeated patterns helps in predicting future sequence values.
• Efficient Calculation: Knowing the common difference allows you to directly calculate any term in the sequence using the formula: \[ a_n = a_1 + (n - 1) imes d \] where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, and \( d \) is the common difference.

These analytical skills are not just limited to sequences but can be applied in various mathematical contexts, emphasizing structure and predictability.

Mathematics Education and Arithmetic Sequences

In mathematics education, understanding arithmetic sequences equips students with foundational analytical skills. These skills are beneficial not only for academics but also for real-life applications such as finance and computing.
Teaching arithmetic sequences supports:

• Critical Thinking: Encouraging students to identify and articulate patterns fosters deeper comprehension.
• Problem Solving: Arithmetic sequences serve as excellent examples for explaining systematic methods of approaching mathematical problems.
• Practical Application: Sequences are found in everyday situations like budgeting weekly expenses or scheduling tasks.

Providing engaging problems and demonstrations helps demystify mathematical concepts, making them accessible and enjoyable. Students gain confidence as they master these basic principles, preparing them for more advanced mathematical exploration.