A Sequence Has A Common Ratio Of 3/2

A Sequence with a Common Ratio of 3/2: Exploring Geometric Progressions

A common ratio of 3/2 signifies a fascinating type of sequence known as a geometric progression (GP). In this in-depth exploration, we'll delve into the characteristics, properties, and applications of geometric progressions with this specific common ratio. We will cover fundamental concepts, explore advanced applications, and equip you with the tools to confidently tackle problems involving such sequences.

Understanding Geometric Progressions

A geometric progression is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). In our case, r = 3/2. This means each subsequent term is 1.5 times the preceding term.

Key characteristics of a GP with r = 3/2:

• Exponential Growth: Because r > 1, the sequence exhibits exponential growth. Each term increases significantly as the sequence progresses. This contrasts with arithmetic progressions, where the increase is linear.
• Predictability: Knowing the first term (a) and the common ratio (r), we can accurately predict any term in the sequence.
• Applications in various fields: Geometric progressions find applications in finance (compound interest), biology (population growth), and physics (radioactive decay—though that would involve a common ratio less than 1).

Formulae for Geometric Progressions

Several key formulas govern geometric progressions. These enable us to easily calculate various aspects of the sequence:

• nth term: The nth term of a GP is given by the formula: a_n = a * r^(n-1), where 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.
• Sum of the first n terms: The sum of the first n terms (S_n) is calculated using: S_n = a * (1 - r^n) / (1 - r) (for r ≠ 1). Note that this formula is crucial for many applications.
• Sum to infinity: For a GP with |r| < 1 (i.e., the common ratio is between -1 and 1), the sum to infinity (S∞) converges to a finite value: S∞ = a / (1 - r). This is particularly useful in scenarios involving infinitely decreasing quantities.

Examples and Illustrations

Let's consider a few examples to solidify our understanding. Suppose the first term of our GP (a) is 2. Then the sequence unfolds as follows:

• a₁ = 2
• a₂ = 2 * (3/2) = 3
• a₃ = 3 * (3/2) = 4.5
• a₄ = 4.5 * (3/2) = 6.75
• a₅ = 6.75 * (3/2) = 10.125

and so on. Observe the consistent 1.5x growth.

Using the formulas:

• Finding the 10th term (a₁₀): a₁₀ = 2 * (3/2)^(10-1) = 2 * (3/2)^9 ≈ 38.44

• Sum of the first 5 terms (S₅): S₅ = 2 * (1 - (3/2)^5) / (1 - (3/2)) ≈ 26.125

Advanced Applications and Problem Solving

The applications of geometric progressions extend beyond simple calculations. Let's explore some real-world scenarios:

1. Compound Interest

Compound interest is a classic example of a geometric progression. Imagine investing $1000 with an annual interest rate of 50%, compounded annually.

• Year 1: $1000 * 1.5 = $1500
• Year 2: $1500 * 1.5 = $2250
• Year 3: $2250 * 1.5 = $3375

This illustrates a GP with a = $1000 and r = 1.5. We can use the formulas to determine the investment's value after any number of years.

2. Population Growth

In idealized scenarios, population growth can be modeled using geometric progressions. If a population increases by 50% each year, the population size forms a geometric sequence with r = 1.5.

3. Radioactive Decay (Adaptation)

While typically represented with r < 1, we can adapt the concept. If we consider the remaining amount of a radioactive substance, the inverse of its decay rate could be considered a common ratio >1 in a reversed perspective.

4. Geometric Series in Calculus

The concept of geometric series plays a crucial role in calculus, particularly in the study of infinite series and convergence. The ability to determine whether a geometric series converges (and its sum if it does) is essential for various mathematical analyses.

Solving Problems Involving Geometric Progressions with r = 3/2

Let's tackle some example problems:

Problem 1: The first term of a geometric progression is 5, and the common ratio is 3/2. Find the 7th term.

Solution: Using the formula a_n = a * r^(n-1), we have:

a₇ = 5 * (3/2)^(7-1) = 5 * (3/2)^6 ≈ 5 * 11.39 ≈ 56.95

Problem 2: The sum of the first four terms of a geometric progression is 17.875, and the common ratio is 3/2. Find the first term.

Solution: We use the sum formula S_n = a * (1 - r^n) / (1 - r). Plugging in the given values:

17.875 = a * (1 - (3/2)^4) / (1 - (3/2)) Solving for 'a', we get a = 2.

Problem 3: A bacteria culture doubles its size every 30 minutes. If the initial size is 1000, what will the size be after 2 hours?

Solution: Since it doubles every 30 minutes, the common ratio is 2. In 2 hours (120 minutes), there are 4 periods of doubling. This forms a GP with a = 1000 and r = 2. The size after 2 hours can be found using a_n = a * r^(n-1):

a₅ = 1000 * 2^(4) = 16000.

Conclusion: Mastering Geometric Progressions with r = 3/2

Understanding geometric progressions with a common ratio of 3/2 empowers you to solve a variety of problems across numerous disciplines. By mastering the fundamental formulas and exploring the applications detailed above, you'll be well-equipped to tackle complex scenarios and gain a deeper appreciation for the power of mathematical sequences. Remember to practice applying these concepts to further strengthen your understanding and problem-solving skills. The consistent 1.5x multiplier might seem simple at first glance, but its implications in various exponential growth scenarios are significant. Remember to always carefully define your terms and correctly apply the relevant formulas to achieve accurate results.