Determining The Relationship Between Successive Terms In The Sequence 2.4, -4.8, 9.6, -19.2

Determining the relationship between successive terms in a sequence is a fundamental concept in mathematics. This article delves into the given sequence, $2.4, -4.8, 9.6, -19.2$, to identify the pattern governing its terms. We will explore the concepts of common difference and common ratio to pinpoint the correct relationship. Understanding sequences is crucial for various mathematical applications, including calculus, discrete mathematics, and even real-world problem-solving. Let's embark on this exploration to enhance our understanding of mathematical sequences.

Understanding Sequences and Their Relationships

In the world of mathematics, a sequence is an ordered list of numbers, often following a specific pattern or rule. Identifying this pattern is key to understanding the relationship between the terms. There are two primary relationships we'll focus on: the common difference and the common ratio. A common difference indicates an arithmetic sequence, where each term is obtained by adding a constant value to the preceding term. On the other hand, a common ratio signifies a geometric sequence, where each term is found by multiplying the previous term by a constant factor. To determine which relationship applies to our sequence, $2.4, -4.8, 9.6, -19.2$, we need to examine the differences and ratios between consecutive terms. This examination will allow us to classify the sequence and understand its underlying structure. This understanding is not only crucial for solving problems related to sequences but also for grasping broader mathematical concepts. Recognizing patterns in sequences builds a strong foundation for advanced mathematical studies.

Analyzing the Sequence: $2.4, -4.8, 9.6, -19.2$

To decipher the relationship between successive terms in the sequence $2.4, -4.8, 9.6, -19.2$, let's first investigate the possibility of a common difference. This involves calculating the difference between consecutive terms. The difference between the second term (-4.8) and the first term (2.4) is -4.8 - 2.4 = -7.2. Now, let's check the difference between the third term (9.6) and the second term (-4.8): 9.6 - (-4.8) = 14.4. Since -7.2 and 14.4 are not equal, there is no common difference in this sequence. This indicates that the sequence is not arithmetic. Next, we'll explore the possibility of a common ratio. To do this, we'll divide each term by its preceding term. The ratio of the second term (-4.8) to the first term (2.4) is -4.8 / 2.4 = -2. The ratio of the third term (9.6) to the second term (-4.8) is 9.6 / -4.8 = -2. Finally, the ratio of the fourth term (-19.2) to the third term (9.6) is -19.2 / 9.6 = -2. Since the ratio between each pair of consecutive terms is consistently -2, we can conclude that the sequence has a common ratio of -2. This confirms that the sequence is geometric.

Evaluating the Options

Now that we've analyzed the sequence and determined that it has a common ratio of -2, let's evaluate the given options:

A. The common difference is -7.2. B. The common difference is -2.4. C. The common ratio is -2.0. D. The common ratio is -0.5.

As we calculated earlier, the difference between the first two terms is -7.2, but the difference between the second and third terms is 14.4. Thus, options A and B, which suggest a common difference, are incorrect. We found that the sequence has a common ratio of -2. Option C accurately states this, while option D suggests a common ratio of -0.5, which is incorrect. Therefore, the correct answer is C. The common ratio is -2.0. This exercise highlights the importance of accurately calculating differences and ratios to identify the pattern in a sequence. By systematically examining the terms, we can confidently determine the relationship between them and select the correct option.

The Significance of Common Ratio in Geometric Sequences

The common ratio is a fundamental characteristic of geometric sequences, playing a crucial role in determining the sequence's behavior and properties. In the sequence $2.4, -4.8, 9.6, -19.2$, the common ratio of -2 dictates that each term is obtained by multiplying the previous term by -2. This multiplication by a negative value results in an alternating sign pattern, where positive and negative terms appear in succession. Understanding the significance of the common ratio allows us to predict future terms in the sequence and analyze its long-term behavior. For instance, if we wanted to find the fifth term, we would simply multiply the fourth term (-19.2) by -2, yielding 38.4. Moreover, the common ratio is instrumental in calculating the sum of a finite or infinite geometric series. A geometric series is the sum of the terms in a geometric sequence, and its convergence or divergence depends on the value of the common ratio. If the absolute value of the common ratio is less than 1, the infinite series converges to a finite sum. Conversely, if the absolute value is greater than or equal to 1, the series diverges. Thus, the common ratio is a powerful tool for understanding and manipulating geometric sequences and series.

Real-World Applications of Sequences

Sequences, particularly arithmetic and geometric sequences, are not just abstract mathematical concepts; they have numerous real-world applications. Understanding these applications can highlight the practical relevance of studying sequences. Arithmetic sequences, with their common difference, are used to model situations involving linear growth or decay. For example, the simple interest earned on a fixed deposit over time forms an arithmetic sequence, as the interest earned each year is constant. Similarly, the depreciation of an asset's value over time, if it depreciates by a fixed amount each year, can be modeled using an arithmetic sequence. Geometric sequences, characterized by their common ratio, are used to model exponential growth or decay. Compound interest, where interest is earned not only on the principal but also on the accumulated interest, is a classic example of a geometric sequence. The growth of a population, the spread of a disease, and the decay of radioactive substances are other phenomena that can be modeled using geometric sequences. Moreover, sequences and series find applications in computer science, particularly in the analysis of algorithms and data structures. The efficiency of an algorithm can often be expressed in terms of a sequence, and understanding the properties of that sequence can help optimize the algorithm. In finance, geometric series are used to calculate the present and future values of annuities and other financial instruments. Thus, the study of sequences provides a valuable framework for understanding and solving problems in various fields.

Conclusion

In summary, determining the relationship between successive terms in a sequence is a fundamental skill in mathematics, with wide-ranging applications. By analyzing the sequence $2.4, -4.8, 9.6, -19.2$, we identified that it has a common ratio of -2, making it a geometric sequence. This understanding allowed us to correctly answer the question and reinforce the importance of distinguishing between arithmetic and geometric sequences. The common difference and common ratio are key indicators of these types of sequences, and their proper calculation is essential for solving related problems. Furthermore, we explored the significance of the common ratio in geometric sequences and its role in determining the sequence's behavior and properties. Finally, we highlighted the real-world applications of sequences, demonstrating their practical relevance in fields such as finance, computer science, and population modeling. By mastering the concepts of sequences and their relationships, students can build a strong foundation for advanced mathematical studies and problem-solving in various domains.