Which Of The Following Sequences Are Geometric? Select Three Options: A. \(-2.7, -9, -30, -100, \ldots\) B. \(-1, 2.5, -6.25, 15.625, \ldots\) C. \(9.1, 9.2, 9.3, 9.4, \ldots\) D. \(8, 0.8, 0.08, 0.008, \ldots\) E. \(4, -4, -12, -20, \ldots\)

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In mathematics, a geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Understanding and identifying geometric sequences is a fundamental concept in algebra and calculus. This article will delve into how to determine whether a given sequence is geometric, using examples to illustrate the process. We'll analyze several sequences, identifying the common ratio and explaining why certain sequences qualify as geometric while others do not. This comprehensive guide will provide you with the tools necessary to confidently identify geometric sequences and understand their underlying structure.

What is a Geometric Sequence?

At its core, a geometric sequence involves a pattern of multiplication. The defining characteristic is the presence of a common ratio, denoted as 'r'. This ratio is the constant factor by which each term is multiplied to obtain the next term in the sequence. To determine if a sequence is geometric, you must ascertain whether there is a consistent ratio between consecutive terms. This involves dividing any term by its preceding term and verifying if the result is constant throughout the sequence. If a constant ratio exists, the sequence is geometric; otherwise, it is not.

Consider a sequence such as 2, 6, 18, 54, ... To check if it’s geometric, we calculate the ratio between consecutive terms:

  • 6 / 2 = 3
  • 18 / 6 = 3
  • 54 / 18 = 3

Since the ratio is consistently 3, this sequence is geometric with a common ratio of 3. The implications of this constant ratio are significant. It allows us to predict future terms in the sequence, derive a general formula for the nth term, and understand the growth or decay behavior of the sequence. Geometric sequences are pervasive in various mathematical and real-world contexts, including compound interest calculations, population growth models, and the analysis of exponential phenomena. Recognizing and understanding them is, therefore, a crucial skill in mathematical literacy.

How to Identify a Geometric Sequence

Identifying a geometric sequence involves a straightforward yet critical process. The primary step is to calculate the ratio between consecutive terms. This is done by dividing each term in the sequence by the term that precedes it. For instance, if you have a sequence denoted as a1, a2, a3, ..., you would calculate the ratios a2/a1, a3/a2, and so on. The essence of a geometric sequence lies in the consistency of this ratio. If the calculated ratio is the same across all pairs of consecutive terms, then the sequence is geometric. This constant ratio is the common ratio, and it is the defining feature of geometric sequences. Understanding this concept is crucial for distinguishing geometric sequences from other types of sequences, such as arithmetic sequences where a common difference is added, or more complex sequences that do not follow a consistent pattern.

To illustrate, consider the sequence 4, 8, 16, 32, ... We calculate the ratios:

  • 8 / 4 = 2
  • 16 / 8 = 2
  • 32 / 16 = 2

Since the ratio is consistently 2, this sequence is geometric. Conversely, if we examine a sequence like 1, 4, 9, 16, ... which represents the squares of natural numbers, the ratios are:

  • 4 / 1 = 4
  • 9 / 4 = 2.25
  • 16 / 9 ≈ 1.78

Here, the ratios are not constant, indicating that the sequence is not geometric. This method provides a clear and direct way to determine the nature of a sequence, and it is a foundational skill in the study of sequences and series.

Analyzing the Given Sequences

To accurately determine which sequences are geometric from the given options, we must apply the method of calculating ratios between consecutive terms. This involves systematically dividing each term by its preceding term and observing whether a constant ratio emerges. This process is crucial because it confirms whether the sequence adheres to the multiplicative pattern characteristic of geometric sequences. For each sequence, we will perform these calculations and analyze the results to arrive at a conclusive answer.

Sequence A: -2.7, -9, -30, -100, ...

Let's calculate the ratios between consecutive terms:

  • -9 / -2.7 ≈ 3.33
  • -30 / -9 ≈ 3.33
  • -100 / -30 ≈ 3.33

The ratio appears to be constant at approximately 3.33. Therefore, sequence A is geometric.

Sequence B: -1, 2.5, -6.25, 15.625, ...

Calculating the ratios:

  • 2.5 / -1 = -2.5
  • -6.25 / 2.5 = -2.5
  • 15.625 / -6.25 = -2.5

The ratio is constant at -2.5, confirming that sequence B is geometric.

Sequence C: 9.1, 9.2, 9.3, 9.4, ...

This sequence is an arithmetic sequence, not a geometric one. We can confirm this by calculating the differences between consecutive terms:

  • 9.2 - 9.1 = 0.1
  • 9.3 - 9.2 = 0.1
  • 9.4 - 9.3 = 0.1

The common difference is 0.1, which characterizes an arithmetic sequence, not a geometric one.

Sequence D: 8, 0.8, 0.08, 0.008, ...

Let's examine the ratios:

  • 0.8 / 8 = 0.1
  • 0.08 / 0.8 = 0.1
  • 0.008 / 0.08 = 0.1

Sequence D exhibits a constant ratio of 0.1, indicating it is a geometric sequence.

Sequence E: 4, -4, -12, -20, ...

Calculating the ratios:

  • -4 / 4 = -1
  • -12 / -4 = 3

The ratios are not constant, so sequence E is not geometric.

Conclusion

In summary, to determine whether a sequence is geometric, it is essential to check for a constant ratio between consecutive terms. By dividing each term by its preceding term, we can identify this ratio. If the ratio is consistent throughout the sequence, the sequence is geometric. Applying this method to the given sequences, we identified sequences A, B, and D as geometric, each with its distinct common ratio. Understanding this fundamental property enables us to analyze and predict the behavior of various sequences, which is crucial in many areas of mathematics and real-world applications.

By calculating the ratios between consecutive terms, we determined that sequences A, B, and D are geometric sequences.

Therefore, the correct options are A, B, and D.