Problem 15 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio.Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio. $$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$$

Step-by-Step Solution

Answer

No, the sequence \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots\) is not geometric as it does not have a constant ratio.

Step 1: Identify the sequence

The sequence in question is \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots\)

Step 2: Inspecting the sequence pattern

For the sequence to be geometric, each term should be a product of the previous term and a fixed number (the common ratio). From the first term to the second, the multiplier seems to be 1/2. But from the second term to the third, the multiplier is 2/3. These are different, thus the sequence does not meet the criteria of a geometric sequence, which needs a constant ratio.

Step 3: Conclusion

Because the ratio is not constant, this sequence is not a geometric sequence, hence, does not have a common ratio.

Key Concepts

Understanding Common Ratio

A common ratio is a fundamental element of a geometric sequence. It is the constant factor you multiply by each term to obtain the next term. This consistency is what distinguishes a geometric sequence from other types of sequences.
For example, in a geometric sequence like \( 2, 4, 8, 16, \ldots \), the common ratio is \( 2 \), because each term is obtained by multiplying the previous term by \( 2 \). Here's how you can identify a common ratio:

• Choose any two consecutive terms in the sequence.
• Divide the second term by the first term.
• If the sequence is geometric, this quotient will be the same for all consecutive terms.

In the original exercise sequence \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \), you do not have a constant quotient, hence, it lacks a common ratio.

Recognizing Sequence Patterns

Sequence patterns are the identifiable rules followed by the elements of a sequence. Observing these patterns helps us classify sequences into types such as arithmetic, geometric, or neither. In a geometric sequence, this pattern involves multiplying each term by a common ratio to get the next one.
When trying to identify sequence patterns, consider these steps:

• Look for a consistent pattern between terms: addition, subtraction, multiplication, or division.
• For geometric sequences, ensure the pattern involves a multiplicative constant.
• If no constant pattern is discernible, the sequence could belong to another category, like arithmetic (constant addition or subtraction) or a non-standard type.

With the sequence \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \), the pattern involves division by increasing integers, which doesn’t hold the consistent multiplication needed for a geometric sequence.

Introduction to Geometric Progression

A geometric progression, also known as a geometric sequence, is characterized by each term being a consistent multiple of the previous one. This qualification is met through a common ratio, which remains unchanged throughout the sequence. Geometric progressions can either increase or decrease, dependent on whether the common ratio is greater or less than one.
Key aspects of geometric progression include:

• The first term, which starts the sequence.
• The common ratio, multiplied with each term to yield the next.
• A predictable growth pattern, be it exponential growth or decay.

A sequence like \( 2, 4, 8, 16, \ldots \) exemplifies a geometric progression with a common ratio of \( 2 \). In contrast, a sequence such as \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \) does not maintain a constant common ratio and thus fails to form a geometric progression.