Geometric Sequence Problem Solving Demonstrating K Satisfies 3k^2-62k+40=0

Introduction to Geometric Sequences

In the realm of mathematics, geometric sequences hold a prominent position, characterized by a constant ratio between consecutive terms. Understanding these sequences is crucial for various applications, from financial calculations to scientific modeling. To grasp the intricacies of geometric sequences, it's essential to delve into their fundamental properties and explore the relationships between their terms.

A geometric sequence, at its core, is a series of numbers where each term is obtained by multiplying the preceding term by a fixed value, known as the common ratio. This consistent ratio is the defining characteristic of geometric sequences and sets them apart from other types of sequences, such as arithmetic sequences, where terms differ by a constant difference. The elegance of geometric sequences lies in their predictable pattern, allowing us to extrapolate future terms based on the initial terms and the common ratio.

Consider a geometric sequence with the first term denoted as a and the common ratio as r. The sequence unfolds as follows: a, ar, ar^2, ar^3, and so on. Each term is simply the product of the previous term and the common ratio. This simple yet powerful concept allows us to express any term in the sequence in terms of the first term and the common ratio. For instance, the nth term of the geometric sequence can be represented as ar^(n-1).

The common ratio, r, plays a pivotal role in determining the behavior of the geometric sequence. If r is greater than 1, the terms of the sequence will increase exponentially, leading to a rapidly growing sequence. Conversely, if r is between 0 and 1, the terms will decrease exponentially, approaching zero as the sequence progresses. A negative value of r introduces an alternating pattern, where the terms alternate between positive and negative values. The magnitude of r still dictates the growth or decay of the terms, but the alternating signs add another layer of complexity.

Geometric sequences are not merely abstract mathematical constructs; they manifest in various real-world scenarios. Compound interest, for example, follows a geometric progression, where the initial investment grows exponentially with each compounding period. Population growth, under ideal conditions, can also be modeled using a geometric sequence, assuming a constant growth rate. In physics, radioactive decay exhibits a geometric pattern, with the amount of radioactive substance decreasing by a constant fraction over time.

To further illustrate the concept, let's consider a practical example. Suppose we have a geometric sequence with the first term being 2 and the common ratio being 3. The sequence would then be 2, 6, 18, 54, and so on. Each term is three times the previous term, showcasing the exponential growth characteristic of geometric sequences with a common ratio greater than 1. If we were to reverse the common ratio, making it 1/3, the sequence would become 2, 2/3, 2/9, 2/27, and so on, demonstrating exponential decay as the terms approach zero.

In this exploration of geometric sequences, we've touched upon the fundamental definition, the significance of the common ratio, and the prevalence of these sequences in real-world phenomena. The ability to identify and analyze geometric sequences is a valuable tool in various fields, enabling us to understand and predict patterns of exponential growth and decay. As we delve deeper into the specific problem at hand, involving the first three terms of a geometric sequence, this foundational understanding will prove invaluable in unraveling the solution.

Problem Statement

The problem at hand presents us with the first three terms of a geometric sequence: 3k + 4, 12 - 3k, and k + 16, where k is a constant. Our primary objective is to demonstrate that k satisfies the equation 3k^2 - 62k + 40 = 0. This involves leveraging the properties of geometric sequences and algebraic manipulation to establish the desired relationship.

To embark on this problem, it's crucial to reiterate the defining characteristic of a geometric sequence: the constant ratio between consecutive terms. In this context, it implies that the ratio between the second term (12 - 3k) and the first term (3k + 4) must be equal to the ratio between the third term (k + 16) and the second term (12 - 3k). Mathematically, this can be expressed as:

(12 - 3k) / (3k + 4) = (k + 16) / (12 - 3k)

This equation forms the cornerstone of our solution. It encapsulates the inherent relationship within the geometric sequence and provides a pathway to determine the possible values of k that satisfy the given conditions. The next step involves algebraic manipulation to simplify this equation and transform it into a more manageable form, ultimately leading us to the quadratic equation we aim to demonstrate.

The process of simplification will involve cross-multiplication, expanding the resulting expressions, and rearranging terms to bring them to one side of the equation. This is a standard algebraic technique used to eliminate fractions and consolidate terms, making the equation easier to analyze and solve. As we proceed with these steps, it's essential to maintain accuracy and attention to detail to avoid errors that could derail our solution.

Furthermore, it's worth noting that the quadratic equation we are aiming to derive (3k^2 - 62k + 40 = 0) is a specific type of polynomial equation. Quadratic equations have a characteristic form of ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable we seek to solve for. In our case, the variable is k, and the coefficients are 3, -62, and 40. The solutions to a quadratic equation can be found using various methods, such as factoring, completing the square, or the quadratic formula.

In the context of this problem, we are not required to solve the quadratic equation for k. Our immediate goal is simply to demonstrate that k satisfies this equation. However, the fact that we are dealing with a quadratic equation implies that there may be two possible values of k that satisfy the conditions of the geometric sequence. This is a common characteristic of quadratic equations, and it's something to keep in mind as we interpret the results of our analysis.

As we move forward, we will meticulously execute the algebraic manipulations, ensuring that each step is logically sound and mathematically correct. The journey from the initial ratio equation to the final quadratic equation requires careful attention to detail and a solid understanding of algebraic principles. By the end of this process, we will have successfully demonstrated that the constant k indeed satisfies the equation 3k^2 - 62k + 40 = 0, thus fulfilling the primary objective of the problem.

Derivation of the Equation

To demonstrate that k satisfies the equation 3k^2 - 62k + 40 = 0, we begin with the fundamental property of geometric sequences: the constant ratio between consecutive terms. As established earlier, this implies:

(12 - 3k) / (3k + 4) = (k + 16) / (12 - 3k)

The first step in simplifying this equation is to eliminate the fractions by cross-multiplying. This involves multiplying both sides of the equation by the denominators of the fractions. Cross-multiplication yields:

(12 - 3k) * (12 - 3k) = (3k + 4) * (k + 16)

Now, we need to expand both sides of the equation. Expanding the left side, we have:

(12 - 3k) * (12 - 3k) = 144 - 36k - 36k + 9k^2 = 9k^2 - 72k + 144

Expanding the right side, we get:

(3k + 4) * (k + 16) = 3k^2 + 48k + 4k + 64 = 3k^2 + 52k + 64

Now, our equation looks like this:

9k^2 - 72k + 144 = 3k^2 + 52k + 64

To proceed, we need to consolidate all the terms on one side of the equation to set it equal to zero. This is achieved by subtracting the terms on the right side from the terms on the left side:

9k^2 - 72k + 144 - (3k^2 + 52k + 64) = 0

Simplifying this expression, we have:

9k^2 - 72k + 144 - 3k^2 - 52k - 64 = 0

Combining like terms, we get:

(9k^2 - 3k^2) + (-72k - 52k) + (144 - 64) = 0

This simplifies to:

6k^2 - 124k + 80 = 0

At this point, we notice that all the coefficients in the equation are divisible by 2. Dividing the entire equation by 2, we obtain a simplified form:

(6k^2 - 124k + 80) / 2 = 0 / 2

This yields:

3k^2 - 62k + 40 = 0

This is precisely the equation we aimed to demonstrate. Through a series of algebraic manipulations, starting from the fundamental property of geometric sequences and employing cross-multiplication, expansion, and simplification, we have successfully shown that the constant k satisfies the equation 3k^2 - 62k + 40 = 0.

Conclusion

In conclusion, we have successfully demonstrated that the constant k, derived from the first three terms of the geometric sequence (3k + 4, 12 - 3k, k + 16), satisfies the quadratic equation 3k^2 - 62k + 40 = 0. This demonstration involved a methodical application of the properties of geometric sequences and algebraic manipulation techniques.

The key to this solution lies in the understanding that in a geometric sequence, the ratio between consecutive terms remains constant. By equating the ratios (12 - 3k) / (3k + 4) and (k + 16) / (12 - 3k), we established a fundamental equation that encapsulated the relationship between the terms of the sequence. This equation served as the starting point for our algebraic journey.

The subsequent steps involved a series of algebraic manipulations, each carefully executed to preserve the integrity of the equation. Cross-multiplication was employed to eliminate fractions, transforming the equation into a more manageable form. Expansion of the resulting products revealed the underlying polynomial structure, paving the way for simplification.

Consolidating like terms and rearranging the equation brought us closer to our desired outcome. The process of moving all terms to one side of the equation and setting it equal to zero is a standard technique in algebra, allowing us to identify and isolate the relationship between the variable k and the coefficients of the equation.

The final step involved simplifying the equation by dividing all coefficients by a common factor, in this case, 2. This yielded the quadratic equation 3k^2 - 62k + 40 = 0, which is precisely what we aimed to demonstrate.

The significance of this result extends beyond the specific problem at hand. It underscores the power of algebraic manipulation in unraveling mathematical relationships. By applying fundamental principles and employing systematic techniques, we can transform complex expressions into simpler, more understandable forms.

Moreover, this problem highlights the interconnectedness of mathematical concepts. Geometric sequences, ratios, and quadratic equations are seemingly disparate topics, yet they converge in this problem, demonstrating the unifying nature of mathematics. The ability to recognize and apply these connections is a hallmark of mathematical proficiency.

While we have successfully demonstrated that k satisfies the given equation, the problem does not require us to solve for the specific values of k. However, the fact that we have arrived at a quadratic equation implies that there are potentially two values of k that would satisfy the conditions of the geometric sequence. Solving for these values would involve further algebraic techniques, such as factoring or the quadratic formula.

In conclusion, this exercise has not only provided us with a solution to a specific problem but has also reinforced our understanding of geometric sequences, algebraic manipulation, and the interconnectedness of mathematical concepts. The journey from the initial problem statement to the final equation has been a testament to the power of logical reasoning and systematic problem-solving.