Finding The Formula For The Sequence 27, 9, 3, 1, 1/3 - A Comprehensive Guide

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In the realm of mathematics, sequences play a pivotal role in understanding patterns and relationships between numbers. A sequence is simply an ordered list of numbers, often following a specific rule or formula. One common type of sequence is a geometric sequence, where each term is obtained by multiplying the previous term by a constant value, known as the common ratio. Today, we embark on a journey to decipher the formula that governs the sequence 27, 9, 3, 1, 1/3, ... This exploration will involve analyzing the sequence, identifying the pattern, and ultimately determining the correct formula from the given options.

Decoding the Sequence: Recognizing the Pattern

To begin our investigation, let's carefully examine the sequence 27, 9, 3, 1, 1/3, ... Our primary objective is to discern the relationship between consecutive terms. Is there a constant value that we can add, subtract, multiply, or divide to get from one term to the next? By observing the sequence, we can notice that each term is smaller than the previous one, suggesting a multiplicative relationship with a factor less than 1.

Let's delve deeper and calculate the ratio between consecutive terms:

  • 9 / 27 = 1/3
  • 3 / 9 = 1/3
  • 1 / 3 = 1/3
  • (1/3) / 1 = 1/3

As we can see, the ratio between any two consecutive terms is consistently 1/3. This observation strongly indicates that the sequence is a geometric sequence with a common ratio of 1/3. Now that we have identified the pattern, we can proceed to evaluate the given formulas and determine which one accurately describes the sequence.

Evaluating the Formulas: Finding the Perfect Match

We are presented with four potential formulas, each expressing the nth term (a_n) in terms of the previous term (a_{n-1}) and the first term (a_1). Let's analyze each option:

A. a_n = -3a_{n-1}; a_1 = 27

This formula suggests that each term is obtained by multiplying the previous term by -3. If we apply this to the first term (a_1 = 27), the second term (a_2) would be -3 * 27 = -81. This clearly contradicts our sequence, where the second term is 9. Therefore, option A is incorrect.

B. a_n = (1/3)a_{n-1}; a_1 = 27

This formula indicates that each term is obtained by multiplying the previous term by 1/3. Let's test this formula:

  • a_2 = (1/3) * a_1 = (1/3) * 27 = 9
  • a_3 = (1/3) * a_2 = (1/3) * 9 = 3
  • a_4 = (1/3) * a_3 = (1/3) * 3 = 1
  • a_5 = (1/3) * a_4 = (1/3) * 1 = 1/3

This formula perfectly generates the given sequence. Therefore, option B appears to be the correct answer. However, let's examine the remaining options to ensure we have the most accurate solution.

C. a_n = a_{n-1} - 3; a_1 = 27

This formula implies that each term is obtained by subtracting 3 from the previous term. Applying this to the first term, the second term would be 27 - 3 = 24, which does not match our sequence. Hence, option C is incorrect.

D. a_n = 3a_{n-1}; a_1 = 27

This formula suggests that each term is obtained by multiplying the previous term by 3. If we apply this to the first term, the second term would be 3 * 27 = 81, which again contradicts our sequence. Therefore, option D is incorrect.

The Verdict: Option B is the Key

After carefully evaluating all the options, we can confidently conclude that option B, a_n = (1/3)a_{n-1}; a_1 = 27, is the formula that accurately describes the sequence 27, 9, 3, 1, 1/3, ... This formula encapsulates the geometric nature of the sequence, where each term is obtained by multiplying the previous term by the common ratio of 1/3.

Understanding Recursive Formulas: A Deeper Dive

The formula we identified, a_n = (1/3)a_{n-1}; a_1 = 27, is a recursive formula. A recursive formula defines a sequence by expressing each term in relation to the preceding term(s). In this case, the nth term (a_n) is defined as one-third of the (n-1)th term (a_{n-1}), with the first term (a_1) explicitly given as 27. Recursive formulas are powerful tools for describing sequences where the pattern depends on previous terms.

To fully grasp the concept of recursive formulas, let's compare them with explicit formulas. An explicit formula, also known as a closed-form formula, defines the nth term directly in terms of n, without relying on previous terms. For instance, an explicit formula for the given sequence could be a_n = 27 * (1/3)^(n-1). This formula allows us to calculate any term in the sequence directly by substituting the value of n.

While both recursive and explicit formulas can describe sequences, they offer different advantages. Recursive formulas are often more intuitive for expressing patterns where the relationship between consecutive terms is clear. Explicit formulas, on the other hand, are more convenient for calculating specific terms in the sequence without having to compute all the preceding terms.

Exploring the Significance of Geometric Sequences

Geometric sequences, like the one we analyzed, are ubiquitous in mathematics and have numerous real-world applications. They appear in various contexts, such as:

  • Compound Interest: The growth of an investment with compound interest follows a geometric sequence, where the common ratio is the interest rate plus 1.
  • Population Growth: In certain scenarios, population growth can be modeled using a geometric sequence, assuming a constant growth rate.
  • Radioactive Decay: The decay of radioactive substances follows an exponential pattern, which can be described using a geometric sequence.
  • Fractals: Geometric sequences play a crucial role in the construction of fractals, intricate geometric shapes with self-similar patterns.

The understanding of geometric sequences and their formulas is essential for solving problems in diverse fields, ranging from finance and biology to physics and computer science.

Conclusion: Mastering the Art of Sequence Identification

In this comprehensive exploration, we successfully identified the formula that describes the sequence 27, 9, 3, 1, 1/3, .... We accomplished this by carefully analyzing the sequence, recognizing the pattern of a geometric sequence with a common ratio of 1/3, and systematically evaluating the given formulas. Our investigation led us to the correct answer: option B, a_n = (1/3)a_{n-1}; a_1 = 27.

Furthermore, we delved into the concept of recursive formulas, contrasting them with explicit formulas and highlighting their respective strengths. We also explored the significance of geometric sequences, underscoring their prevalence in various mathematical and real-world applications. By mastering the art of sequence identification, we equip ourselves with a valuable tool for understanding patterns and relationships in the world around us.

Which formula from the options can describe the sequence 27, 9, 3, 1, 1/3, ...?

Finding the Formula for the Sequence 27, 9, 3, 1, 1/3 - A Comprehensive Guide