Which Of The Following Is A Geometric Sequence

Which of the Following is a Geometric Sequence? A Deep Dive into Geometric Progressions

Identifying a geometric sequence might seem straightforward at first glance, but understanding the nuances and applications of these fascinating mathematical progressions requires a deeper dive. This comprehensive guide will not only equip you with the knowledge to confidently identify geometric sequences but also explore their properties, applications, and related concepts.

What is a Geometric Sequence?

A geometric sequence, also known as a geometric progression (GP), 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 (r). This common ratio is the key characteristic that distinguishes a geometric sequence from other types of sequences, like arithmetic sequences.

For example:

• 2, 4, 8, 16, 32... is a geometric sequence with a common ratio of 2 (each term is multiplied by 2 to get the next term).
• 3, -6, 12, -24, 48... is also a geometric sequence, this time with a common ratio of -2. Note that the common ratio can be negative.
• 100, 10, 1, 0.1, 0.01... is another example, with a common ratio of 0.1 (or 1/10). The common ratio can also be a fraction or decimal.

Key characteristics of a geometric sequence:

• Constant ratio: The defining feature is a constant ratio between consecutive terms.
• Non-zero terms: The common ratio cannot be zero.
• Infinite possibilities: Geometric sequences can be finite (ending after a certain number of terms) or infinite (continuing indefinitely).

Identifying a Geometric Sequence: Step-by-Step Guide

To determine whether a given sequence is geometric, follow these steps:

1. Calculate the ratio between consecutive terms: Choose any two consecutive terms and divide the latter by the former. For example, in the sequence 2, 4, 8, 16..., you'd calculate 4/2 = 2, 8/4 = 2, 16/8 = 2, and so on.

2. Check for consistency: Repeat step 1 for all consecutive term pairs. If the ratio remains constant throughout the sequence, then it's a geometric sequence.

3. Identify the common ratio (r): The constant ratio calculated in steps 1 and 2 is the common ratio (r) of the geometric sequence.

Examples: Identifying Geometric Sequences

Let's analyze a few examples to solidify our understanding:

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

1. 3/1 = 3
2. 9/3 = 3
3. 27/9 = 3
4. 81/27 = 3

The ratio between consecutive terms is consistently 3. Therefore, this is a geometric sequence with a common ratio (r) of 3.

Example 2: Is the sequence 5, 10, 15, 20, 25... a geometric sequence?

1. 10/5 = 2
2. 15/10 = 1.5

The ratio is not constant. Therefore, this is NOT a geometric sequence. This is actually an arithmetic sequence (each term increases by a constant difference of 5).

Example 3: Is the sequence 1, -2, 4, -8, 16... a geometric sequence?

1. -2/1 = -2
2. 4/-2 = -2
3. -8/4 = -2
4. 16/-8 = -2

The ratio is consistently -2. Therefore, this is a geometric sequence with a common ratio (r) of -2. Note the alternating positive and negative signs due to the negative common ratio.

Formulas and Properties of Geometric Sequences

Understanding the formulas associated with geometric sequences is crucial for solving problems and applying them in various contexts.

1. The nth Term Formula:

The nth term (a_n) of a geometric sequence can be calculated using the formula:

a<sub>n</sub> = a<sub>1</sub> * r<sup>(n-1)</sup>

where:

• a_n is the nth term
• a₁ is the first term
• r is the common ratio
• n is the term number

2. The Sum of the First n Terms Formula:

The sum (S_n) of the first n terms of a geometric sequence can be calculated using the formula:

S<sub>n</sub> = a<sub>1</sub> * (1 - r<sup>n</sup>) / (1 - r) (for r ≠ 1)

3. The Sum of an Infinite Geometric Series:

For an infinite geometric series where |r| < 1 (the absolute value of the common ratio is less than 1), the sum (S_∞) converges to a finite value:

S<sub>∞</sub> = a<sub>1</sub> / (1 - r)

Applications of Geometric Sequences

Geometric sequences have numerous applications across various fields, including:

• Finance: Compound interest calculations rely heavily on geometric sequences. The growth of an investment with compound interest follows a geometric progression.
• Biology: Population growth (under ideal conditions) can often be modeled using geometric sequences.
• Physics: Radioactive decay, where the amount of a radioactive substance decreases exponentially, is described by a geometric sequence.
• Computer Science: Recursion and algorithms sometimes involve geometric progressions.

Distinguishing Geometric from Arithmetic Sequences

It's crucial to differentiate geometric sequences from arithmetic sequences. While both are types of sequences, their defining characteristics are distinct:

• Arithmetic Sequence: Each term is obtained by adding a constant difference (d) to the previous term. Example: 2, 5, 8, 11... (common difference d = 3)
• Geometric Sequence: Each term is obtained by multiplying the previous term by a constant ratio (r). Example: 2, 6, 18, 54... (common ratio r = 3)

Advanced Concepts: Convergence and Divergence

The behavior of an infinite geometric series depends on the common ratio (r):

• Convergence (|r| < 1): If the absolute value of the common ratio is less than 1, the sum of the infinite series converges to a finite value (as shown in the sum of an infinite geometric series formula above).
• Divergence (|r| ≥ 1): If the absolute value of the common ratio is greater than or equal to 1, the sum of the infinite series diverges; it does not approach a finite limit. The series will either increase without bound or oscillate without converging.

Conclusion: Mastering Geometric Sequences

Understanding geometric sequences is vital for anyone studying mathematics, finance, or any field involving exponential growth or decay. By mastering the concepts discussed in this article – from identifying geometric sequences to applying their formulas and understanding their applications – you will gain a valuable tool for solving problems and modeling real-world phenomena. Remember the key characteristic: a constant ratio between consecutive terms is the hallmark of a geometric sequence. Practice identifying them, and you'll confidently navigate the world of geometric progressions.