Recursive Formula For Sequence 3, -6, 12, -24, 48...

Hey guys! Today, we're diving into the fascinating world of sequences and recursive formulas. We've got a sequence here: $3, -6, 12, -24, 48, \ldots$, and our mission is to figure out the recursive formula that generates it, given that $f(1) = 3$ and $n \geq 1$. Let's break it down and make it super clear. Think of this as a puzzle – a fun one!

Understanding Recursive Formulas

First, let's get our heads around what a recursive formula actually is. In simple terms, a recursive formula defines the terms of a sequence by relating them to the terms that came before. It's like a set of dominoes falling – each domino's fall is dependent on the one before it. So, instead of having a direct formula to calculate any term (like $f(n) = $ something involving $n$), we have a formula that tells us how to get the next term, $f(n+1)$, if we know the current term, $f(n)$.

To get started with understanding recursive formulas for sequences, we need to recognize that they are a fundamental concept in mathematics, particularly in discrete mathematics and number theory. They provide a powerful way to define sequences where each term is generated based on one or more preceding terms. This is a departure from explicit formulas, where a term can be calculated directly without knowing the previous terms. Recursive formulas, on the other hand, capture the essence of a sequence's pattern by relating consecutive terms. A recursive formula typically consists of two parts: a base case (or cases) and a recursive step. The base case specifies the initial term(s) of the sequence, providing a starting point. For instance, in our given problem, $f(1) = 3$ serves as the base case, telling us the first term of the sequence. The recursive step defines how to calculate the subsequent terms based on the preceding terms. This step is usually expressed as a formula that relates $f(n+1)$ to $f(n)$, or potentially to multiple preceding terms like $f(n)$ and $f(n-1)$, depending on the sequence's pattern. Understanding the base case and the recursive step is crucial for both comprehending and generating a sequence using a recursive formula. Recursive formulas are not just mathematical constructs; they have practical applications in computer science, algorithm design, and various modeling scenarios where patterns evolve iteratively. Recognizing the recursive nature of a problem can often lead to elegant and efficient solutions. For example, in dynamic programming, problems are broken down into overlapping subproblems, and solutions to these subproblems are stored and reused, which aligns perfectly with the concept of recursion. Moreover, recursive thinking is invaluable in understanding complex systems where the state at any given time depends on the previous state(s), such as population growth models, financial market simulations, and network dynamics. Therefore, mastering recursive formulas is an essential skill for anyone delving into advanced mathematical or computational studies.

Analyzing the Given Sequence

Okay, let's look at our sequence: $3, -6, 12, -24, 48, \ldots$. What's happening here? To identify the underlying pattern, let's focus on the relationship between consecutive terms. We need to determine what operation (or operations) transforms one term into the next. Notice that the sequence alternates between positive and negative values, which suggests there's likely a negative factor involved. Also, the magnitude of the numbers is increasing, indicating a multiplicative factor. By observing the sequence, we can see that to get from 3 to -6, we multiply by -2. Similarly, to get from -6 to 12, we again multiply by -2. Continuing this pattern, 12 multiplied by -2 gives -24, and -24 multiplied by -2 gives 48. This consistent multiplication by -2 is a clear indicator of a geometric sequence with a common ratio of -2.

The recognition of a common ratio is a key step in deciphering the recursive formula. In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor, which is the common ratio. This observation simplifies the task of finding the recursive formula because it tells us that the relationship between $f(n+1)$ and $f(n)$ will involve multiplication by this common ratio. The alternating signs in the sequence further confirm the negative nature of the common ratio. This alternating pattern between positive and negative values is a characteristic feature of geometric sequences with a negative common ratio. For instance, if the common ratio were positive, the sequence would either consist of all positive terms or all negative terms (depending on the sign of the initial term). However, the presence of both positive and negative terms clearly points to a negative common ratio. Therefore, identifying the common ratio as -2 is a crucial breakthrough. This common ratio is the key to expressing the recursive relationship. Once we have determined this ratio, we can directly formulate the recursive step of the formula, which will state how to obtain the next term in the sequence by multiplying the current term by -2. This understanding allows us to transition smoothly from observing the sequence's pattern to expressing it mathematically, paving the way for selecting the correct recursive formula from the given options.

Evaluating the Options

Now, let's examine the given options and see which one fits our sequence's behavior:

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

We've already established that each term is multiplied by -2 to get the next term. So, we're looking for a formula that reflects this. Let's test each option.

• Option A: If $f(1) = 3$, then $f(2) = -3 * 3 = -9$. This doesn't match our sequence (-6), so option A is incorrect.
• Option B: If $f(1) = 3$, then $f(2) = 3 * 3 = 9$. This also doesn't match our sequence, so option B is incorrect.
• Option C: If $f(1) = 3$, then $f(2) = -2 * 3 = -6$. This matches our sequence! Let's try one more: $f(3) = -2 * -6 = 12$. This also matches. Bingo!

By meticulously evaluating each option against the observed pattern in the sequence, we systematically eliminate the incorrect choices and pinpoint the correct recursive formula. This process underscores the importance of not only understanding the concept of recursive formulas but also applying a rigorous approach to verify the solution. Each option represents a different potential relationship between consecutive terms, and by substituting values and comparing the results with the actual sequence, we can determine which formula accurately captures the sequence's behavior. This step-by-step evaluation is a hallmark of problem-solving in mathematics, ensuring that the chosen solution is not only plausible but also verifiable. The act of testing each option also reinforces our understanding of how recursive formulas work. We see firsthand how a small change in the formula can lead to a completely different sequence. For instance, the difference between multiplying by -2 (as in the correct option) and multiplying by -3 (as in option A) results in vastly different terms. This sensitivity highlights the precision required when working with recursive formulas and the importance of a methodical approach to ensure accuracy. Moreover, this evaluation process showcases the interplay between observation and calculation. We begin by observing the pattern in the sequence, which guides us in formulating a hypothesis about the recursive relationship. Then, we use calculation to test our hypothesis and either confirm or reject it. This cycle of observation, hypothesis, and verification is a fundamental aspect of the scientific method and is applicable to a wide range of problem-solving scenarios.

The Correct Recursive Formula

Therefore, the correct recursive formula is C. $f(n+1) = -2f(n)$. This formula accurately describes how each term in the sequence is generated by multiplying the previous term by -2.

In summary, finding the correct recursive formula for a sequence involves a few key steps. First, we need to understand what a recursive formula is and how it defines a sequence in terms of its previous terms. Second, we carefully analyze the given sequence to identify the pattern or relationship between consecutive terms. In this case, we recognized the sequence as geometric with a common ratio of -2. Finally, we evaluate the given options by applying them to the sequence and verifying which formula accurately generates the terms. This methodical approach ensures that we arrive at the correct solution with confidence. Recursive formulas are a powerful tool in mathematics, allowing us to define complex sequences in a concise and elegant manner. Mastering the techniques for identifying and working with these formulas is an essential skill for anyone interested in mathematics, computer science, or related fields. The ability to recognize patterns and express them mathematically is a core competency that transcends specific disciplines and enables us to tackle a wide range of problems effectively. Moreover, the process of solving problems involving recursive formulas reinforces critical thinking skills, such as logical reasoning, pattern recognition, and systematic evaluation. These skills are invaluable not only in academic pursuits but also in everyday life, where we constantly encounter situations that require us to analyze patterns, make predictions, and solve problems. Therefore, investing time in understanding and practicing with recursive formulas is a worthwhile endeavor that yields both immediate and long-term benefits.

Final Thoughts

So, there you have it! We've successfully identified the recursive formula for the given sequence. Remember, guys, the key is to break down the problem, analyze the patterns, and test your options. You've got this!