Given The Sequence An=(½)n, Find The Value Of The 8th Term, A⁸.

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In the realm of mathematics, sequences play a pivotal role, offering a structured way to explore patterns and relationships within numbers. Among the various types of sequences, the one defined by the formula An=(½)n holds particular significance. This sequence represents a geometric progression, where each term is obtained by multiplying the previous term by a constant factor. In this comprehensive guide, we will delve into the intricacies of this sequence, focusing on how to find specific terms, particularly A⁸, and unraveling the underlying mathematical concepts.

Understanding the Sequence An=(½)n

At its core, the sequence An=(½)n is a geometric sequence, characterized by a constant ratio between consecutive terms. This ratio, often denoted as 'r', is the cornerstone of geometric sequences, dictating how the sequence progresses. In our case, the ratio 'r' is ½, signifying that each term is half of its predecessor. To grasp the essence of this sequence, let's dissect its components:

  • An: This represents the nth term of the sequence. The subscript 'n' denotes the position of the term within the sequence. For example, A₁ refers to the first term, A₂ to the second term, and so on.
  • (½): This is the constant ratio 'r' that governs the sequence. It signifies that each term is obtained by multiplying the previous term by ½.
  • n: This is the exponent, representing the position of the term in the sequence. It determines how many times the ratio (½) is multiplied by itself.

To illustrate, let's calculate the first few terms of the sequence:

  • A₁ = (½)¹ = ½
  • A₂ = (½)² = ¼
  • A₃ = (½)³ = ⅛
  • A₄ = (½)⁴ = ¹/₁₆

As we progress through the sequence, each term becomes progressively smaller, approaching zero. This behavior is characteristic of geometric sequences with a ratio between 0 and 1.

Delving Deeper: Geometric Sequences and Their Properties

Geometric sequences are a fundamental concept in mathematics, with applications spanning various fields, including finance, physics, and computer science. Understanding their properties is crucial for effectively working with them.

  • General Formula: The general formula for a geometric sequence is An = a * r^(n-1), where 'a' is the first term, 'r' is the common ratio, and 'n' is the term number. In our case, a = ½ and r = ½.
  • Common Ratio: The common ratio 'r' determines the behavior of the sequence. If 'r' is positive and less than 1, the sequence converges towards zero. If 'r' is greater than 1, the sequence diverges towards infinity. If 'r' is negative, the sequence alternates in sign.
  • Sum of a Geometric Series: The sum of the first 'n' terms of a geometric series is given by Sn = a(1 - r^n) / (1 - r). This formula is particularly useful for calculating the sum of a large number of terms without having to individually add them.

Calculating A⁸: Finding the Eighth Term

The core objective of this exploration is to determine the value of A⁸, the eighth term of the sequence An=(½)n. To achieve this, we simply substitute n = 8 into the formula:

A₈ = (½)⁸

Now, let's evaluate this expression:

A₈ = (½) * (½) * (½) * (½) * (½) * (½) * (½) * (½)

A₈ = ¹/₂₅₆

Therefore, the eighth term of the sequence An=(½)n is ¹/₂₅₆. This value further illustrates the diminishing nature of the sequence as it progresses.

Methods for Calculating A⁸: A Comparative Approach

While the direct substitution method is straightforward, let's explore alternative approaches to calculate A⁸, providing a broader perspective on problem-solving:

  • Step-by-Step Calculation: We can calculate A⁸ by iteratively multiplying the previous term by ½. Starting with A₁ = ½, we can find A₂ = ½ * ½ = ¼, A₃ = ¼ * ½ = ⅛, and so on, until we reach A₈. This method, while intuitive, can be time-consuming for larger values of 'n'.
  • Using the General Formula: The general formula for a geometric sequence, An = a * r^(n-1), provides a more efficient way to calculate A⁸. Substituting a = ½, r = ½, and n = 8 into the formula, we get A₈ = ½ * (½)^(8-1) = ½ * (½)⁷ = ¹/₂₅₆. This method directly applies the mathematical principles underlying the sequence.
  • Calculator or Computer: For complex sequences or large values of 'n', utilizing a calculator or computer is the most practical approach. These tools can handle the calculations quickly and accurately, saving time and effort.

Applications of the Sequence An=(½)n

The sequence An=(½)n, while seemingly simple, has applications in various domains. Let's explore some of these practical applications:

  • Radioactive Decay: The decay of radioactive substances follows an exponential pattern, which can be modeled using geometric sequences. The sequence An=(½)n can represent the remaining amount of a radioactive substance after 'n' half-lives.
  • Compound Interest: Compound interest, where interest is earned on both the principal and accumulated interest, can be modeled using geometric sequences. The sequence An=(½)n can represent the growth of an investment over time with a specific interest rate.
  • Computer Science: Geometric sequences find applications in computer science, particularly in algorithms and data structures. For example, the binary search algorithm utilizes the concept of halving the search space, which is related to the sequence An=(½)n.

Real-World Examples: Bringing the Sequence to Life

To further illustrate the applications of the sequence An=(½)n, let's consider some real-world examples:

  • Drug Dosage: In medicine, the concentration of a drug in the bloodstream often decreases exponentially over time. The sequence An=(½)n can model the drug concentration after 'n' time intervals, helping to determine appropriate dosage schedules.
  • Image Compression: Image compression algorithms often employ techniques that reduce the amount of data needed to represent an image. Geometric sequences can be used to model the data reduction process, optimizing compression efficiency.
  • Fractals: Fractals, intricate geometric shapes with self-similar patterns, often exhibit properties related to geometric sequences. The sequence An=(½)n can be used to generate certain types of fractals, showcasing the connection between mathematics and visual art.

Conclusion: Mastering Sequences and Their Applications

In conclusion, the sequence An=(½)n provides a valuable lens through which to understand geometric sequences and their applications. By mastering the concepts and techniques discussed in this guide, you can confidently tackle problems involving sequences and appreciate their significance in various fields. From calculating specific terms like A⁸ to exploring real-world applications, the sequence An=(½)n serves as a testament to the power and versatility of mathematics.

Key takeaways from this exploration include:

  • Understanding the definition and properties of geometric sequences.
  • Calculating specific terms of a sequence using different methods.
  • Recognizing the applications of geometric sequences in various domains.
  • Appreciating the connection between mathematics and the real world.

By embracing these insights, you can enhance your mathematical proficiency and unlock the potential of sequences in problem-solving and beyond.