Which Sequence Has A Common Ratio Of -3

Which Sequence Has a Common Ratio of -3? A Deep Dive into Geometric Progressions

Geometric sequences, a fascinating area within mathematics, are characterized by a constant ratio between consecutive terms. This constant, known as the common ratio, plays a crucial role in defining the sequence's behavior and properties. This article delves into the specifics of geometric sequences with a common ratio of -3, exploring their patterns, formulas, and applications. We'll also tackle how to identify such sequences and solve problems related to them.

Understanding Geometric Sequences and Common Ratios

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, often denoted by 'r'. For example:

• 2, 6, 18, 54, ... (common ratio r = 3)
• 100, 50, 25, 12.5, ... (common ratio r = 0.5)
• -1, 3, -9, 27, ... (common ratio r = -3)

The common ratio is the key to understanding and defining a geometric sequence. It dictates whether the sequence is increasing, decreasing, or alternating in sign. A positive common ratio leads to a sequence where terms maintain the same sign (all positive or all negative), while a negative common ratio results in an alternating sequence (terms alternate between positive and negative).

Geometric Sequences with a Common Ratio of -3: Key Characteristics

When the common ratio (r) is -3, the geometric sequence exhibits a distinctive pattern of alternating signs and exponentially increasing magnitude. Let's analyze this:

• Alternating Signs: Each term alternates between positive and negative. If the first term is positive, the second will be negative, the third positive, and so on. If the first term is negative, the pattern reverses.

• Exponential Growth (in absolute value): Ignoring the signs, the absolute value of the terms grows exponentially. Each term is three times the magnitude of the previous term.

• Formula for the nth term: The general formula for the nth term (a_n) of a geometric sequence is:

  a_n = a₁ * r^(n-1)

  Where:

  • a_n is the nth term
  • a₁ is the first term
  • r is the common ratio (-3 in our case)
  • n is the term number

Let's illustrate this with an example. Consider a geometric sequence with a₁ = 2 and r = -3:

• a₁ = 2
• a₂ = 2 * (-3)^(2-1) = -6
• a₃ = 2 * (-3)^(3-1) = 18
• a₄ = 2 * (-3)^(4-1) = -54
• a₅ = 2 * (-3)^(5-1) = 162

As you can see, the sequence is 2, -6, 18, -54, 162,... This clearly demonstrates the alternating signs and exponential growth.

Identifying Sequences with a Common Ratio of -3

Identifying a geometric sequence with a common ratio of -3 involves checking the ratio between consecutive terms. If the ratio consistently equals -3, you've confirmed the sequence. Let's look at a few examples:

Example 1: -1, 3, -9, 27, -81,...

Here:

• 3 / (-1) = -3
• (-9) / 3 = -3
• 27 / (-9) = -3
• (-81) / 27 = -3

This sequence indeed has a common ratio of -3.

Example 2: 5, -15, 45, -135,...

• (-15) / 5 = -3
• 45 / (-15) = -3
• (-135) / 45 = -3

This is another geometric sequence with a common ratio of -3.

Example 3 (Not a Geometric Sequence): 1, -3, 9, -27, 80...

While the first three terms seem to follow a pattern, the fourth term breaks it. 80/-27 ≠ -3. Therefore, this is not a geometric sequence with r = -3.

Solving Problems Involving Geometric Sequences with r = -3

Many problems involve finding specific terms, sums, or other properties of geometric sequences with a common ratio of -3. Here are some common problem types and how to solve them:

1. Finding a specific term:

Use the formula a_n = a₁ * r^(n-1)

Problem: Find the 10th term of a geometric sequence with a₁ = 4 and r = -3.

Solution:

a₁₀ = 4 * (-3)^(10-1) = 4 * (-3)⁹ = 4 * (-19683) = -78732

2. Finding the sum of the first 'n' terms:

The sum of the first 'n' terms (S_n) of a geometric sequence is given by:

S_n = a₁ * (1 - rⁿ) / (1 - r)

Problem: Find the sum of the first 7 terms of a geometric sequence with a₁ = 1 and r = -3.

Solution:

S₇ = 1 * (1 - (-3)⁷) / (1 - (-3)) = (1 - (-2187)) / 4 = 2188 / 4 = 547

3. Finding the first term given other information:

Sometimes, you're given information about later terms and need to find the first term. You'll still use the general formula, but you might need to solve for a₁.

Problem: The 5th term of a geometric sequence with r = -3 is 81. Find the first term (a₁).

Solution:

We know a₅ = a₁ * (-3)^(5-1) = 81

81 = a₁ * (-3)⁴ 81 = a₁ * 81 a₁ = 1

Applications of Geometric Sequences with r = -3

While seemingly abstract, geometric sequences with a negative common ratio have practical applications in various fields:

• Modeling oscillating systems: In physics and engineering, these sequences can model systems exhibiting damped oscillations, where the amplitude decreases over time but the oscillation continues.

• Financial mathematics: Compound interest calculations with alternating deposits and withdrawals can sometimes be modeled using geometric sequences with negative common ratios.

• Computer science: Certain algorithms and data structures involve sequences with alternating patterns that might be represented by geometric progressions with a negative common ratio.

Conclusion: Mastering Geometric Sequences with r = -3

Understanding geometric sequences, particularly those with a common ratio of -3, is vital for anyone working with mathematical models, patterns, and sequences. The alternating signs and exponential growth (in absolute value) present unique challenges and applications. By grasping the formulas and problem-solving techniques discussed in this article, you'll be well-equipped to tackle a wide range of problems related to these fascinating mathematical structures. Remember the key elements: the alternating pattern of signs, the formula for the nth term, the formula for the sum of the first n terms, and the diverse applications in different fields. Practice is key to mastering these concepts and developing a strong intuition for these sequences.