Solving Recursive Formulas Finding F(2) With F(n) = 0.4f(n-1) + 11

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In the realm of mathematics, recursive formulas play a pivotal role in defining sequences. These formulas provide a method to calculate subsequent terms in a sequence based on the values of preceding terms. This article delves into the intricacies of solving a specific recursive formula problem, providing a comprehensive explanation and addressing frequently asked questions.

Understanding Recursive Formulas

To fully grasp the concept, let’s break down the terminology. A recursive formula is essentially a rule that defines each term in a sequence as a function of the preceding terms. It's like a chain reaction, where the output of one step becomes the input for the next. In simpler terms, to find a specific term, you need to know the term(s) before it. This contrasts with explicit formulas, which directly calculate a term based on its position in the sequence without needing previous terms.

The general structure of a recursive formula involves two key components: the recursive equation itself and the initial condition(s). The recursive equation expresses the relationship between a term and its predecessors. For instance, it might say that the nth term is equal to twice the (n-1)th term plus a constant. The initial condition(s), on the other hand, provide the starting point for the sequence. They define the value(s) of the first term(s), which are necessary to kick-start the recursion. Without these initial conditions, the recursive formula wouldn't have a starting point and couldn't generate a sequence.

Recursive formulas find applications in various areas of mathematics and computer science. They are fundamental to defining sequences like the Fibonacci sequence, where each term is the sum of the two preceding ones. In computer science, recursion is a powerful programming technique where a function calls itself to solve smaller subproblems. Understanding recursive formulas is therefore crucial for anyone delving deeper into these fields. The ability to work with and solve these formulas unlocks a wide range of mathematical and computational possibilities.

Problem Statement

Our primary focus is to determine the second term, denoted as f(2), given the recursive formula f(n) = 0.4 * f(n-1) + 11, and the initial condition f(1) = 4. This means we have a rule that tells us how to find any term in the sequence if we know the term before it, and we are given the first term. The challenge is to use this information to calculate the second term.

The recursive formula, f(n) = 0.4 * f(n-1) + 11, is the heart of the problem. It dictates that to find the value of the nth term, we multiply the value of the (n-1)th term by 0.4 and then add 11. This clearly shows the dependency of each term on its predecessor, highlighting the recursive nature of the sequence. The coefficient 0.4 acts as a scaling factor, while the constant 11 represents a fixed increment. Understanding these components is essential for correctly applying the formula.

The initial condition, f(1) = 4, is the crucial starting point. It provides the value of the first term in the sequence. Without this initial condition, the recursive formula would be incomplete. We wouldn't have a value to plug in to start the calculation. Think of it as the seed that grows into the sequence. This initial value anchors the recursion and allows us to generate subsequent terms. In this case, knowing that the first term is 4 is the key to finding the second term.

The problem essentially asks us to apply the recursive formula once, using the given initial condition. We need to substitute n = 2 into the formula and use the value of f(1) to find f(2). This is a straightforward application of the recursive definition, but it's important to understand the underlying principles to avoid errors. By carefully substituting the values and performing the arithmetic, we can arrive at the solution and determine the value of the second term in the sequence. This exercise demonstrates the power and elegance of recursive formulas in generating sequences from simple rules and initial conditions.

Step-by-Step Solution

Let's embark on a step-by-step solution to determine the value of f(2) using the provided recursive formula and initial condition. This will involve careful substitution and arithmetic calculation to arrive at the correct answer.

  1. Identify the given information:

    • Recursive formula: f(n) = 0.4 * f(n-1) + 11
    • Initial condition: f(1) = 4
    • Target: Find f(2)

    This initial step is crucial for clarity. We have the rule that defines the sequence (f(n) = 0.4 * f(n-1) + 11), the starting point (f(1) = 4), and the specific term we need to find (f(2)). Identifying these pieces of information sets the stage for a focused and accurate solution process. It's like having all the ingredients before you start cooking; you know what you're working with and what you're aiming to achieve.

  2. Substitute n = 2 into the recursive formula:

    To find f(2), we need to replace n with 2 in the recursive formula. This gives us:

    f(2) = 0.4 * f(2-1) + 11

    This substitution is the core of applying the recursive formula. It connects the term we want to find (f(2)) to the preceding term (f(1)), which we already know. By replacing n with 2, we've transformed the general formula into a specific equation that we can solve. It's like zooming in on the particular part of the sequence that we're interested in. The substitution step bridges the gap between the general rule and the specific calculation we need to perform.

  3. Simplify the expression:

    Simplifying the expression, we get:

    f(2) = 0.4 * f(1) + 11

    This simplification step streamlines the equation and makes it clearer what needs to be calculated. The term f(2-1) has been simplified to f(1), highlighting the dependency on the first term. This cleaned-up expression is now ready for the next step, where we will substitute the value of f(1) and perform the arithmetic. It's like organizing your workspace before starting a task; a simplified expression reduces the chances of making errors and makes the calculation more efficient.

  4. Substitute the value of f(1):

    Since we know that f(1) = 4, we can substitute this value into the equation:

    f(2) = 0.4 * 4 + 11

    This substitution brings the initial condition into play. We're now using the given value of the first term to calculate the second term. It's like plugging in a known quantity into an equation to solve for an unknown. By substituting f(1) = 4, we've transformed the equation into a simple arithmetic problem. This step completes the connection between the recursive formula, the initial condition, and the target term, setting us up for the final calculation.

  5. Calculate f(2):

    Now, we perform the arithmetic:

    f(2) = 1.6 + 11

    f(2) = 12.6

    This is the final calculation step. We multiply 0.4 by 4 to get 1.6, and then add 11 to get 12.6. This result is the value of the second term in the sequence, f(2). The arithmetic calculation completes the process, transforming the symbolic equation into a numerical answer. It's the culmination of all the previous steps, where we've carefully applied the recursive formula and initial condition to arrive at the solution.

Therefore, the 2nd term, f(2), is 12.6.

Selecting the Correct Answer

Based on our step-by-step solution, we've determined that f(2) = 12.6. Now, let's analyze the provided options to identify the correct answer:

  • A. f(2) = 11.8
  • B. f(2) = 12.2
  • C. f(2) = 12.6
  • D. f(2) = 13

Comparing our calculated value with the given options, it's clear that option C, f(2) = 12.6, matches our result. Therefore, option C is the correct answer.

This step is crucial for validating our solution. By comparing the calculated value with the provided options, we ensure that we've arrived at the correct answer. It's like checking your work to catch any potential errors. In this case, the exact match between our result and option C confirms the accuracy of our solution process. Selecting the correct answer based on the calculations completes the problem-solving cycle and demonstrates a thorough understanding of the concepts involved.

Conclusion

In conclusion, by applying the recursive formula f(n) = 0.4 * f(n-1) + 11 and the initial condition f(1) = 4, we successfully determined the 2nd term to be f(2) = 12.6. This exercise highlights the importance of understanding and applying recursive formulas in sequence problems. The step-by-step approach ensured accuracy and clarity in the solution process.

Understanding recursive formulas is a valuable skill in mathematics and related fields. It allows us to define sequences in a compact and elegant way, where each term is linked to its predecessors. The ability to work with these formulas opens doors to solving a wide range of problems, from simple arithmetic sequences to more complex patterns. The key to success lies in carefully applying the recursive definition and initial conditions, breaking down the problem into manageable steps, and ensuring accuracy in the calculations.

The problem we solved in this article serves as a foundational example of how recursive formulas work. The process of substituting values, simplifying expressions, and performing arithmetic is a common thread in solving various recursive problems. By mastering these techniques, you can tackle more challenging scenarios and gain a deeper appreciation for the power and beauty of recursive relationships. Recursive thinking is not only essential in mathematics but also in computer science, where it forms the basis for many algorithms and data structures.

FAQ Section

To further solidify your understanding of recursive formulas and the problem-solving process, let's address some frequently asked questions.

Q1: What is a recursive formula?

A recursive formula is a mathematical equation that defines a sequence by relating each term to the preceding terms. It typically consists of a recursive equation and one or more initial conditions. The recursive equation specifies how to calculate a term based on previous terms, while the initial conditions provide the starting point(s) for the sequence. Think of it as a chain reaction, where each link is determined by the links before it. Recursive formulas are powerful tools for describing patterns and relationships in sequences, and they have wide applications in mathematics, computer science, and other fields.

Q2: Why are initial conditions necessary in a recursive formula?

Initial conditions are essential because they provide the starting point for the recursion. Without them, the recursive formula wouldn't have a base to build upon, and the sequence couldn't be generated. Imagine trying to climb a ladder without the first rung; you wouldn't be able to start. The initial conditions anchor the sequence and provide the values needed to kick-start the calculation process. They are like the foundation of a building, providing the necessary support for the structure to grow. In essence, initial conditions are what make a recursive definition complete and meaningful.

Q3: Can all sequences be defined by a recursive formula?

No, not all sequences can be defined by a recursive formula. Some sequences may follow patterns that are better described by explicit formulas, which directly calculate a term based on its position in the sequence. Recursive formulas are most suitable for sequences where each term depends on its predecessors. If there's no clear dependency or if the relationship is complex, a recursive formula might not be the most efficient way to define the sequence. For example, a sequence of prime numbers doesn't have a simple recursive definition, while the Fibonacci sequence is a classic example of a sequence that is naturally defined recursively. The choice between recursive and explicit formulas depends on the nature of the sequence and the pattern it follows.

Q4: How do you solve a recursive formula problem?

The general approach to solving a recursive formula problem involves the following steps:

1.  **Identify the given information:** This includes the recursive formula, the initial condition(s), and the term you need to find.
2.  **Substitute:** Substitute the appropriate value of *n* into the recursive formula to relate the target term to preceding terms.
3.  **Simplify:** Simplify the expression to make the calculation clearer.
4.  **Use initial conditions:** Substitute the given initial condition(s) to obtain numerical values.
5.  **Calculate:** Perform the arithmetic to find the value of the target term.

This step-by-step process ensures a systematic and accurate solution. It's like following a recipe: each step builds upon the previous one to achieve the desired result. By carefully applying the recursive formula and initial conditions, you can unravel the sequence and find the value of any term. Practice and familiarity with these steps will make solving recursive problems a more intuitive and efficient process.

Q5: What are some real-world applications of recursive formulas?

Recursive formulas have numerous applications in various fields. Some examples include:

  • Computer science: Recursion is a fundamental programming technique used in algorithms and data structures, such as tree traversal, sorting algorithms, and fractal generation.
  • Finance: Compound interest calculations can be modeled using recursive formulas.
  • Biology: Population growth models often use recursive relationships.
  • Mathematics: The Fibonacci sequence, a classic example of a recursive sequence, appears in various mathematical contexts and has connections to the golden ratio.
  • Art and nature: Fractal patterns, which exhibit self-similarity at different scales, can be generated using recursive algorithms.

These examples highlight the versatility and power of recursive formulas in modeling real-world phenomena. From the intricate patterns of nature to the complex calculations of finance, recursion provides a framework for understanding and representing processes that evolve over time or across scales. The ability to recognize and apply recursive principles is a valuable asset in many disciplines.

This FAQ section aims to address common questions and provide further insights into recursive formulas. By understanding these concepts, you'll be better equipped to tackle a wider range of problems involving sequences and recursion.