Which Of The Following Sequences Are Geometric

Which of the Following Sequences are Geometric? A Comprehensive Guide

Determining whether a sequence is geometric is a fundamental concept in mathematics, particularly in algebra and pre-calculus. Understanding geometric sequences is crucial for tackling more advanced topics like series, exponential growth, and even some aspects of calculus. This comprehensive guide will not only explain what a geometric sequence is but also provide a robust methodology for identifying them, complete with numerous examples and practice problems.

What is a Geometric Sequence?

A geometric sequence (also known as a geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This common ratio is denoted by 'r'. The general form of a geometric sequence is:

a, ar, ar², ar³, ar⁴, ...

where:

• a is the first term of the sequence.
• r is the common ratio (r ≠ 0).

The key characteristic is the consistent multiplication by the common ratio. Let's look at some examples to solidify this concept.

Examples of Geometric Sequences:

• 2, 6, 18, 54, 162, ... (a = 2, r = 3) Each term is multiplied by 3 to obtain the next.
• 100, 50, 25, 12.5, 6.25, ... (a = 100, r = 0.5) Here, the common ratio is a fraction.
• -1, 2, -4, 8, -16, ... (a = -1, r = -2) Notice that the common ratio can be negative, resulting in alternating signs.
• 1, -1, 1, -1, 1, ... (a = 1, r = -1) This sequence alternates between 1 and -1.

Examples of Sequences that are NOT Geometric:

• 1, 2, 4, 7, 11, ... This is an arithmetic sequence (adding a constant difference).
• 1, 3, 6, 10, 15, ... This is a sequence of triangular numbers; it does not have a constant ratio between consecutive terms.
• 1, 1, 2, 3, 5, ... This is the Fibonacci sequence; it's defined by adding the two preceding terms, not by a constant ratio.
• 2, 4, 8, 16, 30, ... While the first four terms might seem geometric, the presence of '30' breaks the pattern of multiplying by 2.

Identifying Geometric Sequences: A Step-by-Step Approach

To determine if a sequence is geometric, follow these steps:

1. Calculate the ratio between consecutive terms: Divide each term by the preceding term. For example, in the sequence 2, 6, 18, 54..., calculate:

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

2. Check for consistency: If the ratio remains constant throughout the sequence, then you have a geometric sequence. If the ratios are different, it's not geometric.

3. Determine the common ratio (r): The constant ratio calculated in step 1 is your common ratio (r).

Let's apply this methodology to several examples:

Example 1: Is the sequence 3, 9, 27, 81, ... geometric?

1. Ratios: 9/3 = 3; 27/9 = 3; 81/27 = 3
2. Consistency: The ratio is consistently 3.
3. Common Ratio: r = 3

Conclusion: Yes, this is a geometric sequence.

Example 2: Is the sequence 1, 4, 9, 16, ... geometric?

1. Ratios: 4/1 = 4; 9/4 = 2.25; 16/9 ≈ 1.78
2. Consistency: The ratios are not constant.
3. Common Ratio: There is no common ratio.

Conclusion: No, this is not a geometric sequence. This is actually a sequence of perfect squares.

Example 3: Is the sequence 10, 5, 2.5, 1.25, ... geometric?

1. Ratios: 5/10 = 0.5; 2.5/5 = 0.5; 1.25/2.5 = 0.5
2. Consistency: The ratio is consistently 0.5.
3. Common Ratio: r = 0.5

Conclusion: Yes, this is a geometric sequence.

Example 4: Is the sequence -2, 4, -8, 16, -32, ... geometric?

1. Ratios: 4/(-2) = -2; (-8)/4 = -2; 16/(-8) = -2; (-32)/16 = -2
2. Consistency: The ratio is consistently -2.
3. Common Ratio: r = -2

Conclusion: Yes, this is a geometric sequence. Note the alternating signs due to the negative common ratio.

Advanced Considerations and Applications

The identification of geometric sequences extends beyond simple numerical patterns. Understanding geometric sequences is critical in various applications:

1. Financial Modeling:

Compound interest, a cornerstone of financial calculations, relies on geometric sequences. Each year, the interest earned is added to the principal, and the next year's interest is calculated on this larger amount. This creates a geometric progression of the total investment value over time.

2. Population Growth/Decay:

In biology and ecology, populations that grow or decline at a constant percentage rate follow a geometric progression. For instance, bacterial growth in ideal conditions often exhibits this pattern. Similarly, radioactive decay, where a fixed proportion of a substance decays over a given time period, also demonstrates a geometric sequence.

3. Geometric Series:

The sum of the terms of a geometric sequence is called a geometric series. Understanding geometric sequences is essential to finding the sum of these series, which has wide applications in areas such as calculating annuities, determining the total distance traveled by a bouncing ball, and more.

4. Fractals:

Many fractals, complex geometrical shapes with self-similar patterns, are constructed using geometric principles. The recursive nature of fractal generation often involves repeated scaling and transformations that mirror the characteristics of geometric sequences.

Practice Problems:

Test your understanding with these practice problems:

1. Is the sequence 5, 10, 20, 40, 80, ... geometric? If so, what is the common ratio?

2. Is the sequence 1, 1/2, 1/4, 1/8, 1/16, ... geometric? If so, what is the common ratio?

3. Is the sequence 2, 6, 18, 54, 162, ... geometric? If so, what is the common ratio?

4. Is the sequence 1, 0, -1, 0, 1, ... geometric? If so, what is the common ratio?

5. Is the sequence 27, 9, 3, 1, 1/3, ... geometric? If so, what is the common ratio?

6. Is the sequence 100, -50, 25, -12.5, 6.25, ... geometric? If so, what is the common ratio?

7. Is the sequence 1, 2, 4, 8, 16, ... geometric? If so, what is the common ratio?

These examples demonstrate the versatility and practical implications of recognizing and understanding geometric sequences. By mastering this fundamental concept, you build a strong foundation for further mathematical explorations and applications in various fields. Remember, the key is to always check for the consistent common ratio between consecutive terms.