Finding The Recursive Formula For The Sequence 3, -6, 12, -24, 48

In the realm of mathematical sequences, a recursive formula serves as a powerful tool for defining patterns. It establishes a connection between consecutive terms, allowing us to generate a sequence by repeatedly applying the formula. This article delves into the process of identifying the recursive formula that governs a given sequence, using the example sequence 3, -6, 12, -24, 48, where the first term, f(1), is equal to 3. To decipher the recursive nature of this sequence, we must examine the relationship between its terms. We aim to pinpoint a consistent rule that dictates how each term is derived from its predecessor. By unraveling this relationship, we can formulate the recursive formula that accurately represents the sequence's pattern. The challenge lies in recognizing the underlying operation or transformation that connects one term to the next. This could involve addition, subtraction, multiplication, division, or a combination of these operations. Our goal is to express this relationship in a concise mathematical equation, which will then serve as the recursive formula for the sequence.

The sequence provided is: 3, -6, 12, -24, 48. Our mission is to determine which recursive formula can generate this sequence, given that f(1) = 3. To accomplish this, we need to carefully observe the progression of terms and identify the pattern that governs their generation. Let's analyze the relationship between consecutive terms. From the first term (3) to the second term (-6), we notice a change in sign and an increase in magnitude. The second term is -2 times the first term (3 * -2 = -6). Moving from the second term (-6) to the third term (12), we again see a change in sign and an increase in magnitude. The third term is -2 times the second term (-6 * -2 = 12). This pattern continues as we progress through the sequence. The fourth term (-24) is -2 times the third term (12 * -2 = -24), and the fifth term (48) is -2 times the fourth term (-24 * -2 = 48). This consistent multiplication by -2 suggests that the recursive formula involves multiplying the previous term by -2. Now, we can express this relationship mathematically. If we denote the nth term of the sequence as f(n), then the next term, f(n+1), can be obtained by multiplying the previous term, f(n), by -2. This leads us to the recursive formula: f(n+1) = -2 * f(n). This formula encapsulates the essence of the sequence's pattern, where each term is generated by multiplying its predecessor by -2. This concise expression allows us to predict subsequent terms in the sequence and provides a clear understanding of its recursive nature.

Now, let's examine the answer choices provided to determine which one matches the recursive formula we've derived. The answer choices are:

A. f(n+1) = -3 * f(n) B. f(n+1) = 3 * f(n) C. f(n+1) = -2 * f(n) D. f(n+1) = 2 * f(n)

Comparing these options with our derived formula, f(n+1) = -2 * f(n), we can clearly see that option C is the correct match. Option A suggests multiplying the previous term by -3, which does not align with the observed pattern in the sequence. Option B proposes multiplying by 3, which also contradicts the alternating signs in the sequence. Option D suggests multiplying by 2, which, like option B, fails to account for the sign changes. Only option C accurately reflects the relationship between consecutive terms in the sequence, where each term is obtained by multiplying the previous term by -2. This confirms that f(n+1) = -2 * f(n) is the recursive formula that generates the sequence 3, -6, 12, -24, 48. Therefore, the correct answer is option C.

Through careful analysis of the sequence 3, -6, 12, -24, 48, we've successfully identified the recursive formula that governs its pattern. Our analysis revealed that each term is generated by multiplying its predecessor by -2. This led us to the recursive formula f(n+1) = -2 * f(n), where f(n) represents the nth term of the sequence. Upon evaluating the provided answer choices, we confirmed that option C, f(n+1) = -2 * f(n), is the correct match. This formula accurately captures the essence of the sequence's recursive nature, where the alternating signs and doubling magnitude are consistently reflected. The recursive formula provides a concise and elegant way to define the sequence, allowing us to generate any term in the sequence based on its preceding term. In conclusion, the recursive formula f(n+1) = -2 * f(n) is the key to unlocking the pattern within the sequence 3, -6, 12, -24, 48, and option C is the definitive answer.

Recursive formulas are fundamental tools in mathematics, especially when dealing with sequences and series. They provide a way to define a sequence by relating each term to one or more preceding terms. This approach is particularly useful when the sequence follows a pattern that can be expressed in terms of its previous values. To fully grasp the concept of recursive formulas, it's essential to distinguish them from explicit formulas. An explicit formula directly calculates the nth term of a sequence using a formula that depends only on n. In contrast, a recursive formula defines the nth term in relation to previous terms, often the immediately preceding term. This means that to find a specific term using a recursive formula, you need to know the values of the terms that come before it. The process typically starts with one or more initial values, which serve as the foundation for generating subsequent terms. For example, in the sequence we analyzed earlier, f(1) = 3 was the initial value, and the recursive formula f(n+1) = -2 * f(n) allowed us to generate the rest of the terms. Recursive formulas are not just limited to simple arithmetic sequences. They can be used to define complex patterns, such as those found in the Fibonacci sequence, where each term is the sum of the two preceding terms. Understanding recursive formulas is crucial for solving problems involving sequences, series, and other mathematical concepts where patterns are defined in terms of previous values. They provide a powerful way to describe and analyze mathematical relationships that evolve iteratively.

Recursive formulas are not merely theoretical constructs; they have a wide range of practical applications in various fields, including computer science, finance, and physics. In computer science, recursion is a fundamental programming technique where a function calls itself to solve smaller subproblems of the same type. Recursive algorithms are used in various tasks, such as sorting, searching, and traversing data structures like trees and graphs. The efficiency and elegance of recursive solutions often make them the preferred choice for complex problems. In finance, recursive formulas are used to model compound interest, where the interest earned in each period is added to the principal, and the next period's interest is calculated on the new balance. This iterative process can be effectively represented using a recursive formula. The concept of present value and future value also relies on recursive calculations. In physics, recursive relationships can be found in various phenomena, such as the motion of a pendulum or the decay of radioactive substances. The recursive nature of these processes allows physicists to model and predict their behavior over time. Beyond these specific examples, recursive formulas are used in various other areas, including population growth models, fractal geometry, and game theory. Their ability to capture iterative processes and relationships makes them a valuable tool for modeling and analyzing complex systems in diverse fields. Understanding the applications of recursive formulas provides a broader perspective on their significance and highlights their versatility in solving real-world problems.

In conclusion, mastering the art of identifying and applying recursive formulas is an invaluable skill in mathematics and beyond. Recursive formulas provide a powerful means of defining sequences and patterns by relating terms to their predecessors. This approach allows us to capture the essence of iterative processes and relationships, which are prevalent in various fields. Through our analysis of the sequence 3, -6, 12, -24, 48, we successfully demonstrated the process of deriving a recursive formula. By carefully observing the relationship between consecutive terms, we identified the pattern of multiplying by -2 and expressed it concisely as f(n+1) = -2 * f(n). This example highlights the importance of pattern recognition and the ability to translate observations into mathematical expressions. Furthermore, we explored the distinction between recursive and explicit formulas, emphasizing the iterative nature of recursive definitions. We also delved into the diverse applications of recursive formulas, spanning computer science, finance, physics, and more. These applications underscore the practical significance of recursive thinking and its ability to solve complex problems in various domains. As you continue your mathematical journey, embrace the power of recursive formulas. Practice identifying patterns, formulating recursive expressions, and exploring their applications in diverse contexts. By honing your skills in this area, you'll unlock a deeper understanding of mathematical relationships and enhance your problem-solving capabilities. The world of recursive formulas is vast and fascinating, offering endless opportunities for exploration and discovery.