Finding K In A Geometric Sequence Proving 3k^2-62k+40=0

by ADMIN 56 views

In the fascinating world of mathematical sequences, geometric sequences hold a special place. These sequences, characterized by a constant ratio between successive terms, appear in various applications, from financial calculations to physics problems. In this article, we delve into a specific problem involving a geometric sequence, where the first three terms are expressed in terms of a constant, k. Our goal is to demonstrate how to derive a quadratic equation that k must satisfy. This exploration will not only solidify your understanding of geometric sequences but also enhance your problem-solving skills in algebra.

Understanding Geometric Sequences

Before we dive into the problem, let's briefly recap the fundamental concept of geometric sequences. A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric sequence with a common ratio of 3. This is because each term is three times the previous term. Mathematically, if we denote the first term as a and the common ratio as r, then the terms of the geometric sequence can be written as:

  • First term: a
  • Second term: ar
  • Third term: ar²
  • Fourth term: ar³
  • And so on...

The beauty of a geometric sequence lies in its predictable pattern, which allows us to establish relationships between its terms. One crucial relationship is that the ratio between any two consecutive terms is always equal to the common ratio, r. This property is the key to solving many problems involving geometric sequences, including the one we're about to tackle.

The Problem at Hand

Now, let's consider the specific problem we aim to solve. We are given that the first three terms of a geometric sequence are:

  • 3k + 4
  • 12 - 3k
  • k + 16

where k is a constant. Our task is to demonstrate that k satisfies the following quadratic equation:

3k² - 62k + 40 = 0

This problem combines the concept of geometric sequences with algebraic manipulation. To solve it, we'll leverage the property of the common ratio in geometric sequences. We'll set up equations based on the ratios between consecutive terms and then manipulate these equations to arrive at the desired quadratic equation.

Setting Up the Equations: Leveraging the Common Ratio

The cornerstone of solving this problem is the common ratio. In a geometric sequence, the ratio between any two consecutive terms is constant. This means that the ratio between the second term and the first term must be equal to the ratio between the third term and the second term. Mathematically, this can be expressed as:

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

This equation is the heart of our solution. It embodies the fundamental property of geometric sequences and provides a direct link between the given terms and the unknown constant, k. Our next step is to manipulate this equation algebraically to isolate k and eventually arrive at the quadratic equation we're aiming to demonstrate.

Algebraic Manipulation: Unveiling the Quadratic Equation

To transform the equation (12 - 3k) / (3k + 4) = (k + 16) / (12 - 3k) into a more manageable form, we'll begin by cross-multiplying. This eliminates the fractions and allows us to work with a single equation. Cross-multiplication involves multiplying the numerator of the left side by the denominator of the right side, and vice versa. This yields:

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

Now, we need to expand both sides of the equation. This involves multiplying out the terms within the parentheses. On the left side, we have (12 - 3k) multiplied by itself. On the right side, we have the product of two binomials. Expanding these products, we get:

144 - 72k + 9k² = 3k² + 52k + 64

This equation is a significant step forward. We've eliminated the fractions and expanded the products, resulting in a polynomial equation involving k². Our next goal is to rearrange the terms to bring all the terms to one side, setting the equation equal to zero. This will give us the standard form of a quadratic equation.

Rearranging Terms: Standard Form of a Quadratic Equation

To obtain the standard form of a quadratic equation, we need to gather all the terms on one side of the equation, leaving zero on the other side. This involves subtracting the terms on the right side of the equation from both sides. Starting with the equation 144 - 72k + 9k² = 3k² + 52k + 64, we subtract 3k² from both sides, then subtract 52k from both sides, and finally subtract 64 from both sides. This gives us:

144 - 72k + 9k² - 3k² - 52k - 64 = 0

Now, we combine like terms. We combine the k² terms, the k terms, and the constant terms. This simplifies the equation to:

6k² - 124k + 80 = 0

This equation looks very similar to the quadratic equation we are trying to demonstrate. However, to match the target equation exactly, we can simplify it further by dividing all the terms by a common factor.

Simplification: Matching the Target Equation

Observe that all the coefficients in the equation 6k² - 124k + 80 = 0 are divisible by 2. Dividing both sides of the equation by 2, we get:

3k² - 62k + 40 = 0

This is precisely the quadratic equation we set out to demonstrate! We have successfully shown that the constant k in the given geometric sequence satisfies this equation. This result confirms the relationship between the terms of the geometric sequence and provides a crucial piece of information for further analysis, such as finding the possible values of k.

Conclusion: The Power of Geometric Sequences and Algebraic Manipulation

In this exploration, we've successfully navigated a problem involving a geometric sequence and demonstrated that a constant k, which defines the first three terms of the sequence, satisfies a specific quadratic equation. We achieved this by leveraging the fundamental property of geometric sequences – the constant common ratio between consecutive terms. By setting up equations based on this property and then employing algebraic manipulation techniques, such as cross-multiplication, expansion, and simplification, we were able to isolate k and arrive at the desired quadratic equation.

This problem highlights the power of combining concepts from different areas of mathematics. Geometric sequences, with their inherent patterns, provide a rich context for applying algebraic techniques. The ability to translate the properties of geometric sequences into algebraic equations and then manipulate these equations is a valuable skill in problem-solving.

Furthermore, the quadratic equation we derived opens up new avenues for exploration. We can now use various methods, such as factoring, completing the square, or the quadratic formula, to find the possible values of k. These values, in turn, will allow us to determine the specific terms of the geometric sequence. This demonstrates how solving one part of a problem can often lead to further investigations and a deeper understanding of the underlying mathematical concepts.

In conclusion, this journey through geometric sequences and algebraic manipulation underscores the interconnectedness of mathematical ideas and the power of problem-solving techniques. By mastering these concepts and techniques, you'll be well-equipped to tackle a wide range of mathematical challenges and appreciate the elegance and beauty of mathematics.