Finding The Fifth Term Of A Geometric Sequence A Step-by-Step Guide

We are presented with a classic geometric sequence problem. In this problem, we're given that the first term of a geometric sequence is 3, and the third term is 4/3. The core challenge here is to determine the fifth term of this sequence. This requires a solid understanding of the properties of geometric sequences and how to manipulate them to find unknown terms. Geometric sequences, characterized by a constant ratio between consecutive terms, appear frequently in mathematics and have applications in various fields, making it crucial to master the techniques to solve them.

To solve this problem effectively, we will delve into the fundamental properties of geometric sequences. Geometric sequences are sequences of numbers where each term is found by multiplying the previous term by a constant factor, commonly known as the common ratio (r). Identifying this common ratio is the first key step in unraveling the sequence and finding any specific term. Understanding the concept of the common ratio and how it dictates the progression of the sequence is pivotal for tackling problems related to geometric sequences. In this article, we will explore how to extract this common ratio from the given information and use it to calculate the fifth term of the sequence. Furthermore, we will discuss the general formula for the nth term of a geometric sequence, which allows us to find any term directly, given the first term and the common ratio. This formula is a powerful tool in solving various types of geometric sequence problems, and we will illustrate its application in this specific scenario. Finally, we will show how to apply the geometric sequence formula to calculate the fifth term, completing our task. By understanding and applying these principles, you will be well-equipped to solve similar problems involving geometric sequences.

Geometric sequences are the heart of this problem, so let's delve deeper into understanding what they are and how they work. A geometric sequence is a sequence of numbers where each term is multiplied by a constant factor to obtain the next term. This constant factor is termed the common ratio (r). Understanding the structure of geometric sequences is crucial for solving problems related to them. The common ratio plays a central role in determining the progression of the sequence and is the key to unlocking solutions. For instance, if we have a geometric sequence starting with a term 'a' and a common ratio 'r', the sequence unfolds as a, ar, ar², ar³, and so on. Each term is the product of the previous term and the common ratio, forming a clear and predictable pattern. In our problem, we are given two terms of the sequence, which will allow us to determine the common ratio and subsequently find the fifth term. Recognizing the common ratio's significance and its role in shaping the sequence is critical for our approach. By focusing on the relationship between consecutive terms, we can derive a general formula that helps us find any term in the sequence. This understanding is not only useful for this specific problem but also serves as a foundation for tackling more complex geometric sequence problems. In the next sections, we will utilize this understanding to extract the common ratio from the given terms and apply the general formula to determine the fifth term efficiently. Geometric sequences are not just theoretical constructs; they appear in various real-world applications, such as calculating compound interest, modeling population growth, and analyzing radioactive decay. Therefore, mastering geometric sequences has practical implications beyond the classroom.

Finding the common ratio is a crucial step in solving geometric sequence problems. The common ratio, denoted as 'r', is the constant factor by which each term is multiplied to obtain the next term in the sequence. In our problem, we're given the first term (a₁) and the third term (a₃), which provides us with enough information to determine the common ratio. This is because the third term is the first term multiplied by the common ratio squared (a₃ = a₁ * r²). By setting up this equation, we can solve for 'r', which is essential for finding any term in the sequence. The ability to manipulate these relationships between terms is a cornerstone of solving geometric sequence problems. Once we have the common ratio, we can easily move forward to calculate any term in the sequence, including the fifth term, which is our ultimate goal. This process of determining the common ratio is not just applicable to this problem; it is a fundamental technique in handling any geometric sequence scenario where some terms are known and others need to be found. Furthermore, the common ratio can reveal whether the sequence is increasing, decreasing, or oscillating, providing additional insights into its behavior. For instance, if the common ratio is greater than 1, the sequence increases; if it's between 0 and 1, the sequence decreases; and if it's negative, the sequence alternates between positive and negative values. Understanding these implications of the common ratio enhances our ability to analyze and interpret geometric sequences effectively. In the subsequent sections, we will demonstrate how to use the calculated common ratio to find the fifth term of the given sequence.

With the common ratio in hand, we can now use the general formula for a geometric sequence to find the fifth term. The general formula for the nth term (an) of a geometric sequence is given by an = a₁ * r^(n-1), where a₁ is the first term, r is the common ratio, and n is the term number we want to find. This formula is a powerful tool that allows us to directly calculate any term in the sequence without having to list out all the preceding terms. In our case, we want to find the fifth term (a₅), so we will substitute n = 5 into the formula, along with the values we know for a₁ and r. This will give us a direct solution for the fifth term, completing the problem. Applying this formula effectively requires careful substitution and calculation, but it is a much more efficient method than repeatedly multiplying by the common ratio. The general formula encapsulates the essence of geometric sequences, highlighting how each term is related to the first term and the common ratio. By mastering this formula, we gain a versatile tool for solving a wide range of geometric sequence problems. Moreover, understanding the derivation of this formula can provide deeper insights into the structure of geometric sequences and their properties. The exponent (n-1) in the formula reflects the fact that to reach the nth term, we need to multiply the first term by the common ratio (n-1) times. This logical connection between the formula and the underlying structure of the sequence enhances our understanding and problem-solving abilities. In the next step, we will explicitly show how to substitute the values and calculate the fifth term using this formula.

Let's apply what we've learned to solve the problem step-by-step. We're given that the first term (a₁) is 3, and the third term (a₃) is 4/3. Our goal is to find the fifth term (a₅). The first critical step is to determine the common ratio (r). We know that a₃ = a₁ * r², so we can set up the equation 4/3 = 3 * r². Solving for r² involves dividing both sides by 3, giving us r² = 4/9. Taking the square root of both sides, we find that r can be either 2/3 or -2/3. This is an important consideration because a geometric sequence can have a positive or negative common ratio, leading to different patterns in the sequence. Now that we have the possible values for the common ratio, we can use the general formula for the nth term, an = a₁ * r^(n-1), to find the fifth term. For the case where r = 2/3, we substitute a₁ = 3, r = 2/3, and n = 5 into the formula, giving us a₅ = 3 * (2/3)^(5-1) = 3 * (2/3)⁴ = 3 * (16/81) = 16/27. Similarly, for the case where r = -2/3, we substitute a₁ = 3, r = -2/3, and n = 5 into the formula, giving us a₅ = 3 * (-2/3)^(5-1) = 3 * (-2/3)⁴ = 3 * (16/81) = 16/27. Notice that in this particular problem, both possible values of the common ratio lead to the same fifth term. However, this is not always the case, and it's crucial to consider both positive and negative roots when solving for the common ratio. By systematically working through these steps, we have successfully found the fifth term of the geometric sequence.

In conclusion, we have successfully navigated through the problem of finding the fifth term of a geometric sequence. We started by understanding the fundamental properties of geometric sequences, particularly the role of the common ratio. We then determined the common ratio using the given information about the first and third terms. Following this, we applied the general formula for the nth term of a geometric sequence to calculate the fifth term. The problem highlighted the importance of considering both positive and negative roots when solving for the common ratio and demonstrated how the general formula can be efficiently used to find any term in a geometric sequence. By mastering these techniques, you can confidently tackle various problems involving geometric sequences. Furthermore, this problem-solving process reinforces the broader mathematical skills of problem analysis, equation solving, and applying formulas, which are valuable in many areas of mathematics and beyond. Geometric sequences are not just abstract mathematical concepts; they have real-world applications in finance, physics, computer science, and more. Therefore, understanding and being able to work with geometric sequences is a valuable skill. This exercise has not only helped us find a specific term in a sequence but also deepened our understanding of the underlying principles and applications of geometric sequences. This comprehensive approach to problem-solving equips us with the tools to handle similar challenges and further explore the fascinating world of mathematics.