Determining geometric sequences from a set of options requires a clear understanding of what defines a geometric progression. A geometric sequence is a series of numbers where each term is multiplied by a constant value to obtain the next term. This constant value is called the common ratio. In simpler terms, if you divide any term in the sequence by its preceding term, you should get the same ratio throughout the sequence. This principle is the cornerstone for identifying geometric sequences, and by applying it diligently, we can discern whether a given sequence adheres to this mathematical structure. In this comprehensive guide, we will delve into the characteristics of geometric sequences, explore how to identify them, and methodically analyze the given options to pinpoint the correct answer. Understanding geometric sequences is not only crucial for academic success in mathematics but also for various real-world applications, such as calculating compound interest, modeling population growth, and even in the arts, where geometric patterns play a significant role. By mastering the fundamentals of geometric sequences, you will gain a powerful tool for problem-solving and analytical thinking.
Analyzing Option A: 0, 1, 2, 3, ...
To ascertain whether the sequence 0, 1, 2, 3, ... is a geometric sequence, we must examine the ratio between consecutive terms. In a geometric sequence, the ratio between any term and its preceding term must be constant. Let's analyze the given sequence: 0, 1, 2, 3, …. To check if this sequence is geometric, we will divide each term by its preceding term and observe if the ratio is consistent. First, consider the ratio between the second term (1) and the first term (0): 1/0. This is undefined, as division by zero is not permissible in mathematics. This immediately indicates that the sequence cannot be geometric. However, for the sake of completeness, let's examine the ratio between the third term (2) and the second term (1): 2/1 = 2. Next, we look at the ratio between the fourth term (3) and the third term (2): 3/2 = 1.5. Clearly, the ratios between consecutive terms are not constant (undefined, 2, and 1.5), which definitively proves that the sequence 0, 1, 2, 3, ... is not a geometric sequence. This initial assessment highlights a critical aspect of geometric sequences: the common ratio must be consistent throughout the sequence, and the presence of zero as the first term often disrupts this consistency due to the undefined nature of division by zero. Therefore, we can confidently conclude that option A does not represent a geometric sequence. Understanding such nuances is essential for accurately identifying geometric sequences and avoiding common pitfalls in mathematical analysis. The nature of geometric sequences demands a consistent multiplicative relationship between terms, and any deviation from this pattern disqualifies the sequence.
Evaluating Option B: 2, -3, 9/2, -18/4
Let's investigate the sequence 2, -3, 9/2, -18/4 to determine if it qualifies as a geometric sequence. As we established earlier, a geometric sequence is characterized by a constant ratio between consecutive terms. To verify this, we will divide each term by its preceding term and check if the resulting ratios are the same. First, we'll find the ratio between the second term (-3) and the first term (2): -3 / 2 = -1.5. Next, we'll calculate the ratio between the third term (9/2) and the second term (-3): (9/2) / -3 = (9/2) * (-1/3) = -9/6 = -3/2 = -1.5. Finally, we determine the ratio between the fourth term (-18/4) and the third term (9/2): (-18/4) / (9/2) = (-18/4) * (2/9) = -36/36 = -1. Simplifying -18/4 gives us -9/2, so the sequence can also be written as 2, -3, 9/2, -9/2. Rechecking the final ratio, we have (-9/2) / (9/2) = -1. Notice that the ratio between the third and second terms was calculated correctly as -1.5, but the final term -18/4 simplifies to -9/2. The ratio between the fourth and third terms is thus (-9/2) / (9/2) = -1, which is different from -1.5. Therefore, the ratios are -1.5, -1.5, and -1. Since the ratios are not consistent throughout the sequence, the sequence 2, -3, 9/2, -18/4 is not a geometric sequence. This meticulous step-by-step verification process is crucial in accurately identifying geometric sequences. The common ratio is the defining characteristic, and any inconsistency in the ratio disqualifies the sequence from being geometric. The discrepancy in the calculated ratios underscores the importance of careful computation and verification when dealing with mathematical sequences. The process highlights the necessity of double-checking each step to ensure accuracy and avoid misclassifying sequences based on superficial observations.
Examining Option C: 8, 4, 2, 1, 1/2, 1/4, ...
Let's now turn our attention to the sequence 8, 4, 2, 1, 1/2, 1/4, ... to ascertain if it represents a geometric progression. As we've emphasized, a geometric sequence is defined by a constant ratio between successive terms. To verify this, we will divide each term by its preceding term and scrutinize whether the resultant ratio remains consistent throughout the sequence. First, let's compute the ratio between the second term (4) and the first term (8): 4 / 8 = 1/2. Next, we'll calculate the ratio between the third term (2) and the second term (4): 2 / 4 = 1/2. Continuing this process, we find the ratio between the fourth term (1) and the third term (2): 1 / 2 = 1/2. Now, let's look at the ratio between the fifth term (1/2) and the fourth term (1): (1/2) / 1 = 1/2. Lastly, we'll compute the ratio between the sixth term (1/4) and the fifth term (1/2): (1/4) / (1/2) = (1/4) * (2/1) = 2/4 = 1/2. Upon meticulous examination, we observe that the ratio between each consecutive term is consistently 1/2. This uniformity in the ratio definitively establishes that the sequence 8, 4, 2, 1, 1/2, 1/4, ... is indeed a geometric sequence. The common ratio in this geometric sequence is 1/2, indicating that each term is obtained by multiplying the previous term by 1/2. The consistent multiplicative relationship between terms is the hallmark of a geometric sequence, and this sequence exemplifies this characteristic perfectly. Therefore, we can confidently affirm that option C accurately represents a geometric sequence. The ability to identify such sequences hinges on a thorough understanding of the common ratio concept and the systematic verification of its constancy throughout the sequence.
Analyzing Option D: -7, 10, 23, 36, ...
Finally, let's analyze the sequence -7, 10, 23, 36, ... to determine if it is a geometric sequence. As previously discussed, a geometric sequence requires a constant ratio between consecutive terms. To assess this, we will divide each term by its preceding term and check for consistency. First, we'll calculate the ratio between the second term (10) and the first term (-7): 10 / -7 = -10/7, which is approximately -1.4286. Next, we'll find the ratio between the third term (23) and the second term (10): 23 / 10 = 2.3. Immediately, we can see that the ratios -10/7 and 2.3 are not equal. For the sake of thoroughness, let's also calculate the ratio between the fourth term (36) and the third term (23): 36 / 23 ≈ 1.5652. It is evident that the ratios between consecutive terms (-10/7, 2.3, and approximately 1.5652) are not constant. This lack of a common ratio definitively demonstrates that the sequence -7, 10, 23, 36, ... is not a geometric sequence. This analysis underscores the fundamental criterion for identifying geometric sequences: the presence of a consistent ratio between terms. The varying ratios in this sequence indicate that the terms do not follow a multiplicative pattern, which is a prerequisite for geometric progressions. The clear divergence in the calculated ratios serves as conclusive evidence that option D does not represent a geometric sequence. Understanding the importance of a constant ratio is crucial for correctly classifying sequences and distinguishing between arithmetic, geometric, and other types of sequences. The systematic approach of dividing consecutive terms and verifying the consistency of the resulting ratios provides a reliable method for identifying geometric sequences.
Conclusion: The Geometric Sequence Identified
In summary, after a meticulous examination of each option, we have definitively identified the geometric sequence among the given choices. Our analysis focused on the core characteristic of geometric sequences: the presence of a constant ratio between consecutive terms. By systematically dividing each term by its preceding term, we were able to determine whether a common ratio existed throughout the sequence. Option A (0, 1, 2, 3, ...) was immediately disqualified due to the undefined ratio resulting from division by zero and the subsequent inconsistent ratios. Option B (2, -3, 9/2, -18/4) initially showed a constant ratio between the first three terms, but a careful recalculation revealed an inconsistency, leading to its rejection. Option D (-7, 10, 23, 36, ...) exhibited varying ratios from the outset, clearly indicating that it was not a geometric sequence. However, Option C (8, 4, 2, 1, 1/2, 1/4, ...) consistently demonstrated a common ratio of 1/2 between each consecutive term. This uniformity definitively established that option C is a geometric sequence. Therefore, the correct answer is C: 8, 4, 2, 1, 1/2, 1/4, .... This exercise underscores the importance of a thorough and systematic approach when identifying mathematical sequences. The ability to accurately recognize geometric sequences is not only crucial for academic success but also for practical applications in various fields, such as finance, engineering, and computer science. The concept of a common ratio is the cornerstone of geometric sequences, and mastering its application is essential for anyone seeking to understand and work with these mathematical structures effectively. The detailed analysis presented here serves as a comprehensive guide for identifying geometric sequences and highlights the significance of careful computation and verification in mathematical problem-solving.