Problem 12 - Algebra and Trigonometry

Question

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$2,4,6,8, \dots$$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of 2.

Step 1: Understand Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant is known as the 'common difference'.

Step 2: Examine the sequence given

The sequence provided is: 2, 4, 6, 8, ...

Step 3: Find Differences Between Terms

Subtract the first term from the second term: \(4 - 2 = 2\)Subtract the second term from the third term:\(6 - 4 = 2\)Subtract the third term from the fourth term:\(8 - 6 = 2\)

Step 4: Determine if Differences are Constant

The differences calculated in the previous step are all equal to 2. Since the difference between each consecutive term is the same, the sequence is arithmetic.

Step 5: Identify the Common Difference

The common difference of this arithmetic sequence is 2.

Key Concepts

Understanding Common Difference

An arithmetic sequence is defined by its unique characteristic known as the "common difference." This is the consistent amount added to each term in the sequence to generate the next term. For example, if the first term is 3 and the common difference is 5, the sequence would proceed as 3, 8, 13, and so on.
• The formula to find the common difference is: \[ d = a_{n} - a_{n-1} \] where \(d\) is the common difference, \( a_{n} \) is the nth term, and \( a_{n-1} \) is the preceding term.
• In the sequence example given above, the common difference is calculated by subtracting the first term from the second term, second from third, or any term minus its previous term, confirming that the common difference remains constant.
This property is crucial in identifying a sequence as arithmetic, as seen with sequences like 2, 4, 6, and 8, where each number increments by 2.

What are Consecutive Terms?

In mathematics, when we refer to consecutive terms, we talk about terms that appear one after the other in a sequence. In the context of arithmetic sequences, each term is derived by adding the common difference to the preceding one.
Think of them like individual links in a chain. Each link, or term, is attached one directly after the other without gaps. For example, in the sequence 2, 4, 6, 8, each number is a consecutive term.
• To find the difference between consecutive terms: - Simply subtract the previous term from the current term. - If all results are identical, the sequence is confirmed as arithmetic.
Understanding consecutive terms is foundational for grasping the concept of sequences in mathematics and tackling more complex topics involving progressions.

Importance of Mathematics in Sequences

Mathematics plays a crucial role in helping us understand sequences, particularly arithmetic sequences. By applying mathematical reasoning, we can determine the properties of a sequence by its formation rules.
The study of sequences aids in:

• Recognizing patterns and predicting future terms.
• Calculating common differences and sum of terms.
• Understanding more complex mathematical constructs such as series.

The approach described in solving whether a sequence is arithmetic involves basic operations such as addition, subtraction, and the understanding of equality. The skills gained here form the basis for tackling more advanced mathematics.
Engaging with sequences enhances logical thinking and demonstrates the broad application of math in solving real-world problems.