Problem 10 - Algebra 2

Question

Is the given sequence arithmetic? If so, identify the common difference. \(-5,5,-5,5,-5, \dots\)

Step-by-Step Solution

Answer

No, the given sequence is not an arithmetic sequence.

Step 1: Understanding an arithmetic sequence

An arithmetic sequence has a common difference, which is the difference between each successive term in the sequence. This difference is a constant. Therefore, to determine whether a sequence is arithmetic or not, the difference between successive terms must be checked.

Step 2: Check the difference between the terms

Let's find the difference between the successive terms in the sequence: \[ 5 - (-5) = 10, \] \[ -5 - 5 = -10, \] \[ 5 - (-5) = 10, \] And so on, repeating this pattern of 10 and -10.

Step 3: Determine whether the sequence is arithmetic

The difference between the successive terms in the sequence alternates between 10 and -10 instead of being constant. Therefore, the given sequence is not an arithmetic sequence.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial element that dictates the pattern of the sequence. It is the amount by which each term increases or decreases from the previous term. To identify it, subtract any term from the subsequent term. For instance, if you have a sequence like 2, 5, 8, 11, you calculate the common difference as follows:

• Second term minus the first term: 5 - 2 = 3
• Third term minus the second term: 8 - 5 = 3
• Fourth term minus the third term: 11 - 8 = 3

Here, the common difference is 3, demonstrating a consistent step in the sequence. Notably, the common difference remains the same throughout an arithmetic sequence. This clarity allows us to predict the progression of the sequence easily.

Constant Difference

A constant difference is a defining property of arithmetic sequences. It ensures that the difference between consecutive terms remains the same. This consistency is what makes arithmetic sequences predictable and straightforward to analyze. In the case of a legitimate arithmetic sequence, once we calculate the constant difference for one pair of successive terms, it should be identical for every pair in the sequence. Using the example sequence discussed before, 2, 5, 8, 11, we see that no matter which successive pair of terms you consider, the difference is always 3. This constant nature simplifies the sequence, allowing for straightforward progression of terms without complex calculations.

Successive Terms

The concept of successive terms refers to terms that follow one another directly in a sequence. In arithmetic sequences, examining the successive terms is key to understanding the structure of the sequence. By computing the difference between these terms, we can determine if a sequence is arithmetic. In our earlier exercise, the sequence -5, 5, -5, 5 demonstrates alternation rather than consistency between successive terms:

• From -5 to 5, the difference is 10.
• From 5 to -5, the difference is -10.

This alternating pattern indicates that the sequence is not arithmetic, since the successive differences are not identical. For a sequence to be classified as arithmetic, the difference between each successive term must be constant, not alternating.