First Five Terms Of Geometric Sequence First Term 125 Ratio 1/5
In the realm of mathematics, geometric sequences hold a prominent position, characterized by their unique pattern of constant multiplication. Understanding these sequences is crucial for various mathematical applications, from financial calculations to physics problems. This article delves into the process of identifying the first five terms of a geometric sequence, providing a comprehensive explanation and practical examples to solidify your understanding. We will specifically address the case where the first term is 125 and the common ratio is 1/5, walking you through the steps to arrive at the correct sequence. By the end of this exploration, you'll have a firm grasp on how to generate terms in a geometric sequence and confidently tackle similar problems.
Understanding Geometric Sequences
To effectively determine the terms of a geometric sequence, it's essential to first grasp the fundamental concept. A geometric sequence is a series of numbers where each term is obtained by multiplying the preceding term by a constant factor, known as the common ratio. This constant ratio dictates the pattern of growth or decay within the sequence. For instance, if the first term is 2 and the common ratio is 3, the sequence would progress as 2, 6, 18, 54, and so on, with each term being three times the previous one. The common ratio can be positive or negative, leading to either increasing or alternating sequences. Recognizing this multiplicative relationship is key to unraveling the behavior of geometric sequences and predicting their terms.
The formula for the nth term () of a geometric sequence is given by: , where is the first term, r is the common ratio, and n is the term number. This formula provides a direct way to calculate any term in the sequence without having to compute all the preceding terms. Understanding and applying this formula is crucial for efficiently solving problems involving geometric sequences. For example, if you want to find the 10th term of a sequence with a first term of 5 and a common ratio of 2, you would substitute these values into the formula: . This demonstrates the power of the formula in quickly determining specific terms within a geometric progression.
The common ratio, often denoted as r, plays a pivotal role in defining the nature of a geometric sequence. It determines whether the sequence is increasing (if r > 1), decreasing (if 0 < r < 1), or alternating (if r < 0). A common ratio of 1 results in a constant sequence where all terms are the same. Identifying the common ratio is often the first step in analyzing a geometric sequence. To find the common ratio, you can divide any term by its preceding term. For example, in the sequence 2, 6, 18, 54, the common ratio is 6/2 = 18/6 = 54/18 = 3. A clear understanding of the common ratio allows for the accurate prediction and calculation of subsequent terms in the sequence, making it a fundamental element in the study of geometric progressions.
Problem Statement: First Term 125, Common Ratio 1/5
Our specific problem involves a geometric sequence where the first term () is 125 and the common ratio (r) is 1/5. This means we begin with the number 125, and each subsequent term is obtained by multiplying the previous term by 1/5. This scenario represents a decreasing geometric sequence, as the common ratio is a fraction between 0 and 1. The task is to determine the first five terms of this sequence, revealing the pattern of diminishing values as we progress through the series. This problem serves as an excellent example to illustrate the application of the geometric sequence formula and the impact of the common ratio on the sequence's behavior. By solving this, we reinforce the understanding of how geometric sequences operate and how to calculate their terms systematically.
The challenge lies in accurately applying the common ratio repeatedly to generate the subsequent terms. It's crucial to maintain precision in the calculations, especially when dealing with fractions. A small error in any step can lead to an incorrect sequence. To avoid such errors, it's recommended to double-check each multiplication and simplify the fractions whenever possible. The goal is not just to find the correct numbers but also to understand the process of sequential multiplication in a geometric context. This problem is a practical exercise in applying the theoretical knowledge of geometric sequences to a concrete example, enhancing both computational skills and conceptual understanding. Successfully tackling this problem builds confidence in handling geometric sequences and lays the groundwork for more complex mathematical problems.
The significance of this problem extends beyond a simple calculation exercise. It demonstrates a real-world application of geometric sequences, which can be found in various phenomena such as compound interest, population decay, and radioactive decay. Understanding how a quantity changes over time with a constant multiplicative factor is a valuable skill in many fields. This specific example, with a decreasing sequence, can be related to scenarios where something is diminishing by a fixed proportion in each time period. By mastering the mechanics of solving this problem, we gain insights into these broader applications and develop a deeper appreciation for the practical relevance of geometric sequences. The ability to identify and analyze such sequences is a key component of mathematical literacy.
Calculating the First Five Terms
To find the first five terms of the geometric sequence, we start with the given first term, which is 125. The next step is to apply the common ratio, which is 1/5, to this initial term. This means we multiply 125 by 1/5 to get the second term. The result of this multiplication is 25, so the second term in the sequence is 25. This process is then repeated, multiplying each subsequent term by the common ratio to generate the next term in the sequence. This iterative approach forms the core of calculating terms in any geometric sequence, allowing us to move from one term to the next by consistently applying the multiplicative factor. Accuracy in these multiplications is paramount to maintaining the integrity of the sequence.
Next, to find the third term, we multiply the second term (25) by the common ratio (1/5). This gives us 25 * (1/5) = 5. Therefore, the third term in the sequence is 5. Continuing this pattern, we move on to calculate the fourth and fifth terms. For the fourth term, we multiply the third term (5) by the common ratio (1/5), resulting in 5 * (1/5) = 1. So, the fourth term is 1. Finally, to find the fifth term, we multiply the fourth term (1) by the common ratio (1/5), which yields 1 * (1/5) = 1/5. Thus, the fifth term in the sequence is 1/5. This step-by-step calculation demonstrates how each term is derived from the previous one through consistent application of the common ratio, showcasing the fundamental principle of geometric sequences.
By following this iterative process, we have successfully calculated the first five terms of the geometric sequence. These terms are 125, 25, 5, 1, and 1/5. This sequence clearly demonstrates the effect of the common ratio of 1/5, where each term is one-fifth of the previous term. The values decrease steadily, highlighting the nature of a decreasing geometric sequence. This calculated sequence provides a concrete example of how a geometric sequence with a fractional common ratio behaves. The pattern of decreasing values becomes evident as we progress through the terms, offering a visual and numerical representation of the sequence's progression.
Solution and Explanation
Based on our calculations, the first five terms of the geometric sequence, where the first term is 125 and the common ratio is 1/5, are 125, 25, 5, 1, and 1/5. This sequence corresponds to option C in the given choices. Option A incorrectly suggests an arithmetic sequence with a constant addition of 0.2, which does not align with the multiplicative nature of geometric sequences. Option B includes a negative term (-25) and alternates signs, indicating a negative common ratio, which is not the case in our problem. Option D shows an increasing sequence, which contradicts the decreasing nature implied by the common ratio of 1/5. Therefore, option C is the only accurate representation of the geometric sequence described in the problem statement.
The reasoning behind this solution lies in the fundamental definition of a geometric sequence and the correct application of the common ratio. Each term is generated by multiplying the preceding term by the common ratio, which in this case is 1/5. Starting with 125, we successively multiply by 1/5 to obtain the subsequent terms. This process yields 25, 5, 1, and 1/5, confirming the sequence presented in option C. The other options deviate from this pattern, either by using addition instead of multiplication (option A), introducing negative values unnecessarily (option B), or showing an increasing sequence when a decreasing one is expected (option D). Therefore, understanding the core principle of geometric sequences is crucial for arriving at the correct solution.
This solution highlights the importance of accurately applying the definition of a geometric sequence and carefully performing the calculations. It reinforces the concept that each term is a constant multiple of the previous term, as dictated by the common ratio. The process of elimination also plays a role in confirming the correct answer, as we can rule out options that do not adhere to the properties of geometric sequences with a positive fractional common ratio. The problem serves as a practical illustration of how to generate terms in a geometric sequence and how to distinguish it from other types of sequences, such as arithmetic sequences. The correct solution solidifies the understanding of geometric progressions and their characteristics.
Conclusion
In conclusion, we have successfully identified the first five terms of the geometric sequence with a first term of 125 and a common ratio of 1/5. The sequence is 125, 25, 5, 1, and 1/5, which corresponds to option C. This exercise has demonstrated the importance of understanding the fundamental definition of a geometric sequence and the role of the common ratio in determining the terms. By applying the common ratio iteratively, we were able to generate the subsequent terms and arrive at the correct solution. This problem serves as a valuable example of how to apply theoretical knowledge to practical calculations in the realm of mathematics.
The key takeaway from this exploration is the understanding of how a common ratio governs the progression of a geometric sequence. The common ratio acts as the multiplicative factor that links each term to its predecessor, dictating whether the sequence increases, decreases, or alternates. In this specific case, the fractional common ratio of 1/5 resulted in a decreasing sequence, where each term is smaller than the one before it. This concept is crucial for solving various problems involving geometric sequences and for recognizing their patterns in real-world scenarios. The ability to calculate terms in a geometric sequence is a fundamental skill in mathematics, with applications ranging from finance to physics.
Ultimately, mastering the concept of geometric sequences provides a foundation for more advanced mathematical topics and problem-solving. The systematic approach of applying the common ratio to generate terms is a valuable skill that can be transferred to other areas of mathematics and beyond. This example, with its clear and concise solution, serves as a stepping stone for further exploration of sequences and series, fostering a deeper appreciation for the beauty and power of mathematics. The ability to confidently handle geometric sequences is a testament to a strong mathematical foundation and opens doors to a broader understanding of mathematical principles.
