Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-8,\) determine if each sequence is an arithmetic sequence. If the sequence is arithmetic, find the common difference. $$ -3 i,-1 i, 1 i, 3 i, 5 i, \dots $$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of \(2i\).

Step 1: Identify the Terms of the Sequence

The given sequence is \(-3i, -1i, 1i, 3i, 5i, \ldots\). This sequence consists of terms that are imaginary numbers with coefficients changing by a particular pattern.

Step 2: Calculate the Differences Between Consecutive Terms

To determine if the sequence is arithmetic, find the difference between each consecutive pair of terms. Calculate the differences as follows:1. \(-1i - (-3i) = 2i\)2. \(1i - (-1i) = 2i\)3. \(3i - 1i = 2i\)4. \(5i - 3i = 2i\)

Step 3: Check for a Constant Difference

An arithmetic sequence has a constant difference between consecutive terms. In this case, each difference calculated in Step 2 is \(2i\). Thus, the sequence has a common difference of \(2i\) and is arithmetic.

Step 4: Conclusion

Since the difference between consecutive terms is consistent, the sequence is indeed an arithmetic sequence with a common difference of \(2i\).

Key Concepts

Understanding Common Difference in Arithmetic Sequences

In an arithmetic sequence, the term "common difference" refers to the constant amount that separates each term from the next one in the sequence. This difference is uniform across the sequence, meaning every time you move from one term to the next, you add or subtract the same value.

In the exercise provided, the sequence of terms is \(-3i, -1i, 1i, 3i, 5i, \ldots\). To find the common difference, you subtract each term from the one that follows it. For instance, \((-1i) - (-3i) = 2i\). This process is repeated for all terms, confirming a consistent \(2i\) difference.

Key Points:

• The common difference can be positive, negative, or even zero.
• This difference dictates the linear growth or decline of the sequence.
• Recognizing an arithmetic sequence involves verifying that this difference remains constant.

The Role of Imaginary Numbers

Imaginary numbers expand the concept of numbers beyond the real number line. These numbers include the unit \(i\), which represents the square root of \(-1\). In arithmetic sequences involving imaginary numbers, terms are constructed with imaginary components.

In our example, all terms are multiples of \(i\): \(-3i, -1i, 1i, \ldots\). Even though these terms involve "imaginary" parts, the process for determining if there's a common difference remains the same. You treat \(i\) as a variable during calculations just like any other number.

Overview:

• Imaginary numbers are key tools in complex number theory.
• They function alongside real numbers to form complex numbers, written as \(a + bi\).
• Sequences involving imaginary numbers operate under the same arithmetic principles.

Decoding Sequence Patterns

Every sequence is defined by its unique pattern. In arithmetic sequences, this pattern arises from the constant difference between terms. Recognizing these sequences is all about spotting such regularity over their length.

Our example shows a clear sequence pattern: each term progresses by an addition of \(2i\). This steadily growing "pattern" indicates it's an arithmetic sequence. As such, understanding sequence patterns is all about observation and analysis.

Main Ideas:

• Arithmetic sequences are identified by a linear pattern.
• Sequence patterns can involve both real and imaginary numbers.
• Being able to decode these patterns is crucial in math and beyond.