Divergence Of Geometric Sequence Analysis Of 1, -3, 9, -27, 81, ...

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In mathematics, understanding the behavior of sequences is crucial. Sequences can either converge, approaching a specific limit, or diverge, meaning they do not approach a single value. This article delves into the sequence 1, -3, 9, -27, 81, ... to determine whether it converges or diverges. We will explore the characteristics of this sequence, apply relevant mathematical concepts, and provide a clear explanation of why it behaves the way it does. Specifically, we aim to answer the question: Which statement accurately describes this sequence?

Identifying the Sequence Type

To begin, let’s identify the type of sequence we are dealing with. Examining the sequence 1, -3, 9, -27, 81, ..., we notice that each term is obtained by multiplying the previous term by a constant factor. This indicates that we are dealing with a geometric sequence. A geometric sequence is defined as a sequence in which each term is multiplied by a constant value to get the next term. This constant value is known as the common ratio, often denoted as r.

In our given sequence, we can find the common ratio by dividing any term by its preceding term. For instance:

  • -3 / 1 = -3
  • 9 / -3 = -3
  • -27 / 9 = -3
  • 81 / -27 = -3

Thus, the common ratio r for this sequence is -3. Recognizing this, we can express the general form of the sequence as:

an = a1 * r(n-1)

Where:

  • an is the nth term of the sequence,
  • a1 is the first term (which is 1 in our case),
  • r is the common ratio (-3 in our case),
  • n is the term number.

Understanding that this sequence is geometric and has a common ratio of -3 is crucial for determining its convergence or divergence. The common ratio plays a pivotal role in the sequence's behavior as n approaches infinity. Geometric sequences behave differently based on the value of the common ratio, and this behavior dictates whether the sequence converges or diverges. The first term of the sequence is 1, and the common ratio is -3. This means that each subsequent term is obtained by multiplying the previous term by -3. Let's look at the first few terms:

  1. First term: 1
  2. Second term: 1 * (-3) = -3
  3. Third term: -3 * (-3) = 9
  4. Fourth term: 9 * (-3) = -27
  5. Fifth term: -27 * (-3) = 81

As we continue the sequence, the terms will alternate in sign (positive, negative, positive, etc.) and their absolute values will increase exponentially. This pattern is a key indicator of divergence.

Convergence vs. Divergence: The Core Concepts

Before we proceed further, let’s clarify the concepts of convergence and divergence in the context of sequences. A sequence is said to converge if its terms approach a specific, finite value as the term number n approaches infinity. In other words, as we go further and further along the sequence, the terms get closer and closer to a single, fixed number. Mathematically, we can express this as:

limn→∞ an = L

Where L is the limit of the sequence.

On the other hand, a sequence diverges if its terms do not approach a single, finite value as n approaches infinity. This can happen in several ways:

  1. The terms increase or decrease without bound (approaching positive or negative infinity).
  2. The terms oscillate between two or more values without settling on a single limit.

Understanding these definitions is essential for analyzing the behavior of our given sequence. The concept of a limit is central to determining whether a sequence converges or diverges. The limit, if it exists, represents the value that the terms of the sequence approach as n gets larger and larger. For a sequence to converge, this limit must be a finite number. If the terms of the sequence do not approach a finite number, the sequence diverges.

Analyzing Geometric Sequences for Convergence and Divergence

For geometric sequences, the convergence or divergence behavior is primarily determined by the value of the common ratio r. There are specific rules that govern this:

  1. If |r| < 1 (the absolute value of r is less than 1), the sequence converges to 0.
  2. If |r| > 1 (the absolute value of r is greater than 1), the sequence diverges.
  3. If r = 1, the sequence is constant (all terms are the same), and it converges to the first term.
  4. If r = -1, the sequence oscillates between two values and diverges.

Applying these rules to our sequence, where r = -3, we find that |r| = |-3| = 3, which is greater than 1. This immediately tells us that the sequence diverges. The absolute value of the common ratio being greater than 1 means that the terms will increase in magnitude as the sequence progresses. However, the fact that the common ratio is negative adds another layer to the behavior of the sequence: the terms will alternate in sign. This alternation, combined with the increasing magnitude, prevents the sequence from converging to any single value. Instead, the terms jump between large positive and negative values, further confirming its divergent nature.

Applying the Divergence Rule to Our Sequence

In our sequence 1, -3, 9, -27, 81, ..., the common ratio r is -3. Since the absolute value of r is |-3| = 3, which is greater than 1, we can conclude that the sequence diverges. The terms of the sequence alternate in sign and grow in magnitude, preventing the sequence from approaching a single, finite value. This behavior is characteristic of a divergent geometric sequence with |r| > 1.

To further illustrate this, let’s consider the first few terms:

  • 1
  • -3
  • 9
  • -27
  • 81
  • -243
  • 729
  • -2187

As you can see, the terms are not approaching any specific number. They oscillate between large positive and negative values, and the magnitude of these values continues to increase. This pattern confirms that the sequence is indeed divergent.

Conclusion: Determining the Correct Statement

Based on our analysis, the correct statement that describes the sequence 1, -3, 9, -27, 81, ... is:

A. The sequence diverges.

This conclusion is supported by the fact that the sequence is geometric with a common ratio of -3, and the absolute value of the common ratio is greater than 1. This condition guarantees that the terms of the sequence will not converge to a single, finite value. Instead, they will oscillate and grow in magnitude, leading to divergence. Understanding the properties of geometric sequences and the conditions for convergence and divergence is essential for accurately analyzing the behavior of such sequences.

In summary, we have thoroughly examined the sequence 1, -3, 9, -27, 81, ... and determined that it diverges. This determination was made by identifying the sequence as geometric, calculating its common ratio, and applying the rules governing the convergence and divergence of geometric sequences. The analysis provides a clear and concise explanation of why the sequence diverges, solidifying the understanding of this mathematical concept.