Determine The Range Of Geometric Sequence An = {105, 110.25, 115.76, ...}
When exploring mathematical sequences, understanding the range of a geometric sequence is crucial. This article delves into determining the range of a specific geometric sequence, providing a clear explanation and a step-by-step approach to solving this type of problem. Let’s consider the sequence a_n = {105, 110.25, 115.76, ...}. Our goal is to accurately identify the range of this sequence. To do this effectively, it’s important to first have a firm understanding of what a geometric sequence is and how its range is defined. A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. The range, in mathematical terms, refers to the set of all possible output values (terms) of the sequence. It is not merely a set of natural numbers or all real numbers, but rather the specific set of values generated by the sequence itself. Understanding this distinction is key to correctly identifying the range in any given geometric sequence problem. The initial step in determining the range involves examining the sequence and identifying the pattern. This includes calculating the common ratio and observing how the terms progress. In our example, we'll see how each term is derived from the previous one, which will help us define the specific set of numbers that constitute the range. This detailed analysis will guide us in accurately defining the range for the sequence a_n, avoiding common misconceptions and providing a solid foundation for understanding geometric sequence ranges in general. By breaking down the concept and applying it to a specific example, we aim to make the process clear and understandable for anyone studying mathematical sequences.
Identifying the Geometric Sequence
To identify the geometric sequence, we first need to confirm that the given sequence is indeed geometric. This involves verifying that there is a common ratio between consecutive terms. In our case, the sequence is a_n = {105, 110.25, 115.76, ...}. To find the common ratio, we divide any term by its preceding term. Let’s divide the second term by the first term: 110.25 / 105 = 1.05. Now, we'll verify this ratio by dividing the third term by the second term: 115.76 / 110.25 ≈ 1.05. Since the ratio is consistent, we can confidently say that this is a geometric sequence with a common ratio of 1.05. This common ratio is the key to understanding how the sequence progresses and how its terms are generated. Understanding the common ratio is essential because it allows us to predict future terms in the sequence. In a geometric sequence, each term is the product of the previous term and the common ratio. This pattern defines the structure of the sequence and helps us in determining its properties, including its range. Once we've established the common ratio, we can express the nth term of the sequence using the formula a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number. This formula is a powerful tool for analyzing geometric sequences as it provides a general expression for any term in the sequence. In our example, a_1 = 105 and r = 1.05, so the formula becomes a_n = 105 * (1.05)^(n-1). This formula will be instrumental in understanding the range of the sequence, as it shows how the terms increase as n increases. By identifying the geometric sequence and determining its common ratio, we lay the groundwork for accurately defining its range, which is the set of all possible values the sequence can take.
Determining the Range
Determining the range of a geometric sequence involves understanding what values the sequence can take. In the case of the sequence a_n = 105, 110.25, 115.76, ...}, we have already established that it is a geometric sequence with a common ratio of 1.05. This means that each term is 1.05 times the previous term. The first term is 105, and as we multiply by 1.05 repeatedly, the terms will increase. Since we start with 105 and multiply by a factor greater than 1, the terms will continue to grow, but they will remain specific values determined by this multiplication. This is a crucial point. This set accurately represents all the possible values of the sequence. Understanding this concept is vital for distinguishing between different types of sequences and their ranges. In a geometric sequence, the range is a specific set of numbers that follow a particular pattern, rather than a broad category like all real numbers or natural numbers. By correctly identifying the range, we demonstrate a clear understanding of the sequence's behavior and its mathematical properties.
The Correct Answer and Why
The correct answer to the question of the range of the geometric sequence a_n = {105, 110.25, 115.76, ...} is B. The range is {105, 110.25, 115.76, ...}. This is because the range of a sequence is defined as the set of all possible values that the sequence can take. In this case, the sequence starts at 105 and each subsequent term is obtained by multiplying the previous term by the common ratio of 1.05. Therefore, the range consists of the specific numbers generated by this process. Option A, “The range is the set of natural numbers,” is incorrect because the terms of the sequence are not natural numbers (except for the first term) and the sequence does not generate all natural numbers. The terms include decimal values, and the sequence only produces numbers that are 105 multiplied by powers of 1.05. Option C, “The range is all real numbers,” is also incorrect. While the terms are real numbers, the sequence does not produce all real numbers. The terms are limited to those that can be generated by the geometric progression starting with 105 and multiplying by 1.05. Real numbers include a vast, continuous set of values, whereas the range of this sequence is a discrete set of specific values. Understanding why option B is correct involves recognizing that the range is precisely the set of terms that the sequence generates. The sequence is defined by its initial term and common ratio, and these parameters dictate the specific values in the range. This concept is fundamental to understanding sequences and their properties. By selecting the correct answer, we demonstrate an understanding of the definition of a range and how it applies to geometric sequences, highlighting the importance of precise mathematical reasoning.
Common Mistakes to Avoid
When working with geometric sequences, there are common mistakes to avoid when determining the range. One frequent error is assuming that the range is the set of natural numbers. As we’ve seen in the sequence a_n = {105, 110.25, 115.76, ...}, the terms are not all natural numbers. The presence of decimal values indicates that the range is more specific than just the set of positive integers. Another common mistake is thinking that the range of a geometric sequence is all real numbers. While the terms of the sequence are real numbers, the range is not continuous and all-encompassing. The range consists only of the specific values generated by the sequence, which are discrete and follow the pattern defined by the common ratio. Confusing the range with the domain is also a pitfall. The domain of a sequence typically refers to the set of input values (e.g., the term number n), while the range refers to the set of output values (the terms of the sequence). Misunderstanding this distinction can lead to incorrect answers. Additionally, failing to correctly calculate the common ratio can result in misidentifying the pattern of the sequence and, consequently, misrepresenting its range. It's crucial to accurately determine the common ratio by dividing a term by its preceding term and verifying that this ratio is consistent throughout the sequence. To avoid these mistakes, it’s essential to have a clear understanding of the definition of the range and how it relates to geometric sequences. Remembering that the range is the set of all possible values the sequence can take, and carefully analyzing the sequence's pattern and terms, will help in accurately determining the range and avoiding common errors. This careful approach ensures a solid grasp of the concepts and leads to correct solutions.
Conclusion
In conclusion, determining the range of a geometric sequence requires a clear understanding of the sequence's properties and the definition of range. For the sequence a_n = 105, 110.25, 115.76, ...}, we have shown that the range is the specific set of values generated by the sequence itself. This is because the range is the set of all possible output values, and in a geometric sequence, these values are determined by the initial term and the common ratio. The sequence we analyzed has a common ratio of 1.05, which means each term is 1.05 times the previous term. This pattern dictates the specific values that the sequence can take. It's crucial to avoid common mistakes, such as assuming the range is the set of natural numbers or all real numbers. These misconceptions arise from not fully considering the specific nature of the sequence. The range is not a broad category of numbers but rather a precise set of values that the sequence generates. Understanding the difference between the domain and range is also essential. The domain refers to the input values (term numbers), while the range refers to the output values (terms of the sequence). By carefully analyzing the sequence, calculating the common ratio, and applying the correct definition of range, we can accurately determine the set of values that the sequence can take. This process demonstrates a solid grasp of geometric sequences and their properties. In summary, identifying the range involves recognizing the specific pattern of the sequence and avoiding common misconceptions, leading to a clear and accurate understanding of the sequence's behavior.
