Find The Explicit Function For The Sequence 11, 18, 25, ... Is It Arithmetic Or Geometric? If Geometric, Write The Function F(n) =

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Let's embark on a journey to decipher the explicit function that governs the arithmetic sequence 11, 18, 25, ... In the realm of mathematics, sequences play a pivotal role, and understanding their underlying patterns is crucial. This exploration delves into the intricacies of arithmetic sequences, geometric sequences, and the methods to derive their explicit functions. We will dissect the given sequence, identify its defining characteristics, and then construct the explicit function that elegantly captures its behavior. By the end of this exploration, you will gain a profound understanding of how to represent arithmetic sequences using explicit functions, empowering you to predict any term in the sequence without the need for recursive calculations.

Decoding Arithmetic Sequences and Explicit Functions

Arithmetic sequences, the cornerstone of our investigation, are characterized by a constant difference between consecutive terms. This consistent increment, known as the common difference, forms the very essence of an arithmetic sequence's predictable nature. To grasp the concept of explicit functions, envision them as formulas that directly compute any term in a sequence based on its position. Unlike recursive formulas, which rely on preceding terms, explicit functions provide a streamlined pathway to any term, making them invaluable tools for sequence analysis.

The given sequence, 11, 18, 25, ..., immediately hints at an arithmetic progression. The unwavering difference of 7 between successive terms (18 - 11 = 7, 25 - 18 = 7) firmly establishes its arithmetic nature. Our mission now is to translate this observed pattern into a concrete explicit function. The general form of an explicit function for an arithmetic sequence is elegantly expressed as:

  • f(n) = a + (n - 1)d

where:

  • f(n) represents the nth term in the sequence,
  • a denotes the first term,
  • n signifies the term's position in the sequence, and
  • d embodies the common difference.

With this framework in mind, we can seamlessly adapt the general form to our specific sequence. The first term (a) is clearly 11, and the common difference (d) we've already determined to be 7. Substituting these values into the general formula, we arrive at the explicit function that defines our sequence:

  • f(n) = 11 + (n - 1)7

This function stands as a testament to the sequence's pattern, encapsulating its essence in a concise mathematical expression. To further refine our function, we can distribute the 7 and simplify the expression:

  • f(n) = 11 + 7n - 7
  • f(n) = 7n + 4

This simplified form, f(n) = 7n + 4, is the explicit function that governs the arithmetic sequence 11, 18, 25, .... It allows us to directly calculate any term in the sequence by simply plugging in its position (n). For instance, to find the 10th term, we substitute n = 10:

  • f(10) = 7(10) + 4 = 74

This elegant function serves as a powerful tool, enabling us to predict and analyze the sequence's behavior with ease.

Ruling Out Geometric Sequences

Having successfully derived the explicit function for the arithmetic sequence, let's briefly address the possibility of it being a geometric sequence. Geometric sequences, in contrast to arithmetic sequences, are defined by a constant ratio between consecutive terms. To ascertain if our sequence is geometric, we would need to check if the ratio between successive terms remains consistent. Dividing the second term by the first term (18 / 11) yields approximately 1.64. Similarly, dividing the third term by the second term (25 / 18) gives us roughly 1.39. The discrepancy in these ratios definitively rules out the possibility of a geometric relationship. Therefore, our focus remains firmly on the arithmetic nature of the sequence.

Delving Deeper into Explicit Functions and Their Applications

Explicit functions, like the one we've derived, are more than just mathematical curiosities; they are powerful tools with far-reaching applications. Imagine needing to determine the 100th term in the sequence. Using a recursive formula would necessitate calculating all the preceding 99 terms, a tedious and time-consuming task. However, with the explicit function f(n) = 7n + 4, we can directly compute the 100th term:

  • f(100) = 7(100) + 4 = 704

This exemplifies the efficiency and elegance of explicit functions in handling large-scale sequence analysis. Beyond individual term calculations, explicit functions serve as the foundation for various mathematical operations. They can be used to determine the sum of a finite number of terms in an arithmetic sequence, find the average of a set of terms, and even model real-world phenomena that exhibit arithmetic growth. For example, the simple interest earned on a fixed deposit account follows an arithmetic progression, and an explicit function can accurately predict the interest earned over any period.

The Significance of Understanding Sequence Patterns

Understanding sequence patterns, whether arithmetic or geometric, is a fundamental skill in mathematics and beyond. Sequences appear in diverse contexts, from financial modeling to computer science algorithms. The ability to identify patterns, derive explicit functions, and make predictions is invaluable in problem-solving and decision-making. For instance, in computer science, sequences are used to analyze the efficiency of algorithms, with explicit functions representing the time complexity as the input size grows. In finance, understanding compound interest involves geometric sequences, and explicit functions are crucial for projecting investment growth.

The process of deriving an explicit function from a sequence is not merely a mechanical exercise; it is an exercise in mathematical reasoning. It requires us to observe, analyze, and generalize patterns. This skill is transferable to various domains, fostering critical thinking and analytical abilities. Whether you are a student grappling with mathematical concepts or a professional tackling real-world challenges, the ability to decipher sequence patterns will undoubtedly prove to be a valuable asset.

Conclusion: The Explicit Function Unveiled

In conclusion, we have successfully unraveled the explicit function that defines the arithmetic sequence 11, 18, 25, .... Through careful analysis and application of the general form for arithmetic sequences, we arrived at the elegant function f(n) = 7n + 4. This function encapsulates the sequence's essence, enabling us to calculate any term directly and efficiently. We also explored the significance of explicit functions in broader mathematical contexts and highlighted their diverse applications. Understanding sequence patterns and their explicit representations is a cornerstone of mathematical proficiency, empowering us to analyze, predict, and solve problems across various domains. The journey of deciphering this arithmetic sequence has not only provided us with a specific solution but has also illuminated the power and beauty of mathematical reasoning.