What Is The Common Ratio Of The Sequence 6 54

What is the Common Ratio of the Sequence 6, 54...?

Understanding geometric sequences is fundamental in mathematics, and a key component of that understanding is grasping the concept of the common ratio. This article will delve deep into the concept of common ratio, specifically addressing the sequence 6, 54... and extending the principles to a broader understanding of geometric progressions. We'll explore how to calculate the common ratio, its significance in various applications, and how to solve related 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 constant throughout the sequence. For instance, in the sequence 2, 6, 18, 54..., the common ratio is 3 because each term is obtained by multiplying the preceding term by 3.

The general form of a geometric sequence is represented as:

a, ar, ar², ar³, ar⁴, ...

where:

• a is the first term of the sequence
• r is the common ratio
• arⁿ⁻¹ represents the nth term of the sequence

Calculating the Common Ratio

The common ratio (r) is calculated by dividing any term in the sequence by the preceding term. Therefore, if we have a sequence represented by a₁, a₂, a₃, a₄,..., the common ratio can be found using the following formula:

r = a₂ / a₁ = a₃ / a₂ = a₄ / a₃ = ...

Finding the Common Ratio of the Sequence 6, 54...

Let's apply this knowledge to the sequence 6, 54... We only have two terms, but that's enough to determine the common ratio.

Using the formula above:

r = 54 / 6 = 9

Therefore, the common ratio of the sequence 6, 54... is 9.

Significance of the Common Ratio

The common ratio is not merely a mathematical curiosity; it holds significant importance in various fields:

1. Predicting Future Terms:

Knowing the common ratio allows us to predict subsequent terms in the geometric sequence. For our sequence 6, 54..., the next term would be 54 * 9 = 486, followed by 486 * 9 = 4374, and so on. This predictive power is useful in various applications, such as financial modeling (compound interest) and population growth estimations.

2. Calculating the Sum of a Geometric Series:

A geometric series is the sum of the terms of a geometric sequence. The common ratio plays a crucial role in calculating this sum. The formula for the sum of the first 'n' terms of a geometric series is:

Sₙ = a(1 - rⁿ) / (1 - r) where 'r' ≠ 1

This formula is invaluable in scenarios such as calculating the total accumulated amount in a savings account with compound interest or the total distance traveled by a bouncing ball.

3. Solving Exponential Growth and Decay Problems:

Geometric sequences are intrinsically linked to exponential growth and decay. The common ratio represents the growth or decay factor. A common ratio greater than 1 indicates exponential growth (as in our 6, 54... sequence), while a common ratio between 0 and 1 represents exponential decay. These concepts are applied in numerous fields, including:

• Finance: Compound interest calculations
• Biology: Population growth and decay models
• Physics: Radioactive decay
• Engineering: Signal processing and attenuation

Exploring Variations and Extensions

While we've focused on a simple sequence, understanding common ratios extends to more complex scenarios:

1. Infinite Geometric Series:

If the absolute value of the common ratio (|r|) is less than 1, the geometric series converges to a finite sum, even if it has infinitely many terms. The formula for the sum of an infinite geometric series is:

S∞ = a / (1 - r) (where |r| < 1)

This formula has applications in areas such as calculating the total distance a bouncing ball travels before coming to rest.

2. Geometric Sequences with Negative Common Ratios:

Geometric sequences can have negative common ratios. For instance, the sequence 2, -6, 18, -54... has a common ratio of -3. The terms alternate between positive and negative values. The formulas for the sum of a finite or infinite geometric series still apply, but you must carefully consider the sign of the common ratio.

3. Applications in Financial Mathematics:

Understanding common ratios is crucial for comprehending compound interest. The interest earned in each period is added to the principal, and the interest for the next period is calculated on the new, larger principal. This creates a geometric sequence where the common ratio is (1 + interest rate). This concept is fundamental to understanding loan amortization, investment growth, and other financial calculations.

Real-World Applications and Examples

Let's examine some real-world examples that illustrate the application of common ratios and geometric sequences:

Example 1: Compound Interest:

Imagine you invest $1000 at an annual interest rate of 5%, compounded annually. The balance after each year forms a geometric sequence:

• Year 1: $1000 * 1.05 = $1050
• Year 2: $1050 * 1.05 = $1102.50
• Year 3: $1102.50 * 1.05 = $1157.63

The common ratio is 1.05, representing the annual growth factor.

Example 2: Population Growth:

Suppose a bacterial colony doubles in size every hour. If the initial population is 1000 bacteria, the population after each hour forms a geometric sequence:

• Hour 1: 1000 * 2 = 2000
• Hour 2: 2000 * 2 = 4000
• Hour 3: 4000 * 2 = 8000

The common ratio is 2, representing the doubling factor.

Example 3: Radioactive Decay:

A radioactive substance decays at a rate of 10% per year. If the initial amount is 100 grams, the amount remaining after each year forms a geometric sequence:

• Year 1: 100 * 0.9 = 90 grams
• Year 2: 90 * 0.9 = 81 grams
• Year 3: 81 * 0.9 = 72.9 grams

The common ratio is 0.9, representing the decay factor.

Conclusion

The common ratio is a fundamental concept in the study of geometric sequences and series. Understanding its calculation, significance, and applications in various fields is crucial for solving problems related to exponential growth and decay, financial modeling, and numerous other areas. While we've explored the concept primarily through numerical examples, the underlying principles remain consistent across diverse applications. By mastering the concept of the common ratio, you equip yourself with a powerful tool for tackling complex mathematical and real-world challenges. Remember, the key takeaway is that the common ratio is the constant multiplier that defines the relationship between consecutive terms in a geometric sequence, making it a cornerstone of understanding and predicting the behavior of these sequences.