Determining The Sequence Generated By The Function F(n+1) = F(n) - 2

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In the realm of mathematics, sequences play a fundamental role, offering a structured way to explore patterns and relationships between numbers. Among the diverse types of sequences, those defined by recursive functions hold a special significance. These functions, which express the next term in a sequence based on the preceding one, provide a concise yet powerful means of generating intricate numerical patterns. In this article, we embark on a journey to unravel the sequence generated by the recursive function f(n+1) = f(n) - 2, given the initial condition f(1) = 10. We will delve into the step-by-step process of calculating the terms of this sequence, revealing the underlying pattern and ultimately identifying the correct sequence from a set of options.

Understanding Recursive Functions

Before we embark on our exploration, let's take a moment to understand the essence of recursive functions. In simple terms, a recursive function is a function that calls itself within its own definition. This self-referential nature allows us to define sequences in a compact and elegant manner. In our case, the function f(n+1) = f(n) - 2 exemplifies this concept. It states that to find the *(n+1)*th term of the sequence, we simply subtract 2 from the nth term. This iterative process, starting from a known initial term, allows us to generate the entire sequence.

Decoding the Sequence: A Step-by-Step Approach

Now, let's put our understanding of recursive functions into practice and decipher the sequence generated by f(n+1) = f(n) - 2 with the initial condition f(1) = 10. We will meticulously calculate the first few terms of the sequence, revealing the pattern that governs its progression.

  1. The First Term: We are given that f(1) = 10, which serves as our starting point.
  2. The Second Term: Using the recursive formula, we can calculate f(2) as follows: f(2) = f(1) - 2 = 10 - 2 = 8
  3. The Third Term: Continuing this process, we find f(3): f(3) = f(2) - 2 = 8 - 2 = 6
  4. The Fourth Term: We proceed to calculate f(4): f(4) = f(3) - 2 = 6 - 2 = 4
  5. The Fifth Term: Finally, we determine f(5): f(5) = f(4) - 2 = 4 - 2 = 2

As we meticulously calculate the terms, a clear pattern emerges. Each term is obtained by subtracting 2 from the previous term. This consistent decrement of 2 points towards an arithmetic sequence, where the difference between consecutive terms remains constant.

Unveiling the Pattern: An Arithmetic Sequence

Our calculations have unveiled the underlying pattern of the sequence: it is an arithmetic sequence with a common difference of -2. This means that each term is obtained by adding -2 to the previous term. The sequence starts with 10 and progresses as follows: 10, 8, 6, 4, 2, and so on.

To solidify our understanding, let's delve deeper into the characteristics of arithmetic sequences. An arithmetic sequence is a sequence where the difference between consecutive terms remains constant. This constant difference is known as the common difference, often denoted by d. In our case, the common difference d is -2.

The general form of an arithmetic sequence can be expressed as:

a, a + d, a + 2d, a + 3d, ...

where a represents the first term and d is the common difference.

In our sequence, a = 10 and d = -2. Substituting these values into the general form, we obtain:

10, 10 + (-2), 10 + 2(-2), 10 + 3(-2), ...

which simplifies to:

10, 8, 6, 4, ...

This confirms our earlier observation that the sequence generated by f(n+1) = f(n) - 2 with f(1) = 10 is indeed an arithmetic sequence with a common difference of -2.

Identifying the Correct Sequence

Now that we have a firm grasp of the sequence's pattern, let's turn our attention to the given options and identify the one that matches our findings.

The options presented are:

A. -10, -12, -14, -16, -18, ... B. -2, 8, 18, 28, 38, ... C. 8, 18, 28, 38, 48, ... D. 10, 8, 6, 4, 2, ...

Comparing these options with the sequence we derived (10, 8, 6, 4, 2, ...), it becomes evident that option D perfectly matches our result. Therefore, the correct sequence generated by the function f(n+1) = f(n) - 2 for f(1) = 10 is:

D. 10, 8, 6, 4, 2, ...

Conclusion: A Journey Through Recursive Sequences

In this exploration, we have successfully unraveled the sequence generated by the recursive function f(n+1) = f(n) - 2 with the initial condition f(1) = 10. By meticulously calculating the terms and recognizing the underlying pattern, we identified the sequence as an arithmetic sequence with a common difference of -2. This journey has not only provided us with the correct answer but has also deepened our understanding of recursive functions and arithmetic sequences.

Recursive functions serve as a powerful tool for defining sequences in mathematics, offering a concise way to express intricate patterns. By understanding the principles of recursion and the characteristics of different sequence types, we can effectively analyze and decipher a wide range of mathematical sequences. As we continue our exploration of mathematics, recursive functions and sequences will undoubtedly play a crucial role in unraveling the mysteries of numbers and their relationships.

This exploration highlights the beauty and elegance of mathematics, where seemingly complex patterns can be deciphered through systematic analysis and a deep understanding of fundamental concepts. As we continue our mathematical journey, let us embrace the power of recursive functions and sequences, allowing them to guide us towards new discoveries and insights.

Additional Insights and Applications

The concept of recursive sequences extends far beyond the realm of theoretical mathematics. It finds practical applications in various fields, including computer science, finance, and physics. For instance, recursive functions are extensively used in programming to solve problems that can be broken down into smaller, self-similar subproblems. In finance, recursive models are employed to predict stock prices and analyze investment strategies. In physics, recursive equations are used to describe the behavior of complex systems, such as the motion of a pendulum or the flow of fluids.

Furthermore, the study of recursive sequences provides a foundation for understanding more advanced mathematical concepts, such as difference equations and dynamical systems. Difference equations are recursive equations that relate the values of a function at different discrete time points. Dynamical systems, on the other hand, are mathematical models that describe the evolution of a system over time, often involving recursive relationships.

By mastering the fundamentals of recursive sequences, we equip ourselves with a valuable toolset for tackling a wide range of problems in various disciplines. This knowledge empowers us to analyze complex systems, make informed decisions, and contribute to advancements in science and technology.

As we conclude this exploration, let us remember that mathematics is not merely a collection of formulas and equations. It is a language, a way of thinking, and a powerful tool for understanding the world around us. By embracing the beauty and elegance of mathematics, we unlock our potential to solve problems, innovate, and make a positive impact on society.