Geometric Sequences Finding K And Convergence Analysis
In mathematics, geometric sequences hold a special place due to their predictable nature and wide range of applications. These sequences are characterized by a constant ratio between consecutive terms, a property that allows us to explore interesting relationships and solve various problems. In this article, we delve into a specific geometric sequence problem, focusing on determining the value of a constant 'k' and analyzing the convergence of the sequence.
The problem presents us with the first three terms of a geometric sequence: 3k + 4, 12 - 3k, and k + 16. Our primary goal is to demonstrate that 'k' satisfies the quadratic equation 3k² - 62k + 40 = 0. To achieve this, we will leverage the fundamental property of geometric sequences: the ratio between consecutive terms remains constant. This constant ratio, often denoted as 'r', is the cornerstone of our solution.
Understanding Geometric Sequences and Common Ratio
Before diving into the algebraic manipulations, let's solidify our understanding 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, the common ratio 'r'. For instance, the sequence 2, 6, 18, 54... is a geometric sequence with a common ratio of 3. Each term is three times the previous term.
In our given problem, the sequence 3k + 4, 12 - 3k, k + 16 is geometric. This implies 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, we can express this as:
(12 - 3k) / (3k + 4) = (k + 16) / (12 - 3k)
This equation forms the basis for our journey to find the value of 'k'.
Deriving the Quadratic Equation
Now, let's embark on the algebraic journey to demonstrate that 'k' satisfies the equation 3k² - 62k + 40 = 0. We start with the equation derived from the constant ratio property:
(12 - 3k) / (3k + 4) = (k + 16) / (12 - 3k)
To eliminate the fractions, we cross-multiply:
(12 - 3k)(12 - 3k) = (k + 16)(3k + 4)
Expanding both sides of the equation, we get:
144 - 72k + 9k² = 3k² + 52k + 64
Now, we rearrange the terms to bring everything to one side, setting the equation equal to zero:
9k² - 3k² - 72k - 52k + 144 - 64 = 0
Simplifying the equation, we arrive at:
6k² - 124k + 80 = 0
To further simplify, we can divide the entire equation by 2:
3k² - 62k + 40 = 0
This is the very quadratic equation we aimed to demonstrate. We have successfully shown that 'k' satisfies the equation 3k² - 62k + 40 = 0, based on the fundamental property of constant ratios in geometric sequences.
Solving the Quadratic Equation
The next step in our exploration is to solve the quadratic equation 3k² - 62k + 40 = 0. This will provide us with the possible values of 'k' that satisfy the given conditions. There are several methods to solve quadratic equations, including factoring, completing the square, and the quadratic formula. For this particular equation, we will utilize the quadratic formula, which is a general solution applicable to any quadratic equation in the form ax² + bx + c = 0.
The quadratic formula is given by:
k = [-b ± √(b² - 4ac)] / 2a
In our case, a = 3, b = -62, and c = 40. Plugging these values into the quadratic formula, we get:
k = [62 ± √((-62)² - 4 * 3 * 40)] / (2 * 3)
k = [62 ± √(3844 - 480)] / 6
k = [62 ± √3364] / 6
k = [62 ± 58] / 6
This gives us two possible solutions for 'k':
k₁ = (62 + 58) / 6 = 120 / 6 = 20
k₂ = (62 - 58) / 6 = 4 / 6 = 2/3
Therefore, the two possible values for 'k' are 20 and 2/3. These values will be crucial as we proceed to analyze the convergence of the geometric sequence.
With the possible values of 'k' determined, we now shift our focus to the convergence of the geometric sequence. A geometric sequence converges if its terms approach a finite limit as the number of terms increases infinitely. This convergence is directly linked to the common ratio 'r' of the sequence. A geometric sequence converges if and only if the absolute value of the common ratio is less than 1 (|r| < 1). If |r| ≥ 1, the sequence diverges, meaning its terms either grow infinitely large or oscillate without approaching a limit.
To analyze the convergence of our given sequence, we need to determine the common ratio 'r' for each value of 'k' we found earlier (k = 20 and k = 2/3). The common ratio is calculated by dividing any term by its preceding term. We can use the first two terms of the sequence to calculate 'r':
r = (12 - 3k) / (3k + 4)
Analyzing Convergence for k = 20
Let's substitute k = 20 into the expression for 'r':
r = (12 - 3 * 20) / (3 * 20 + 4)
r = (12 - 60) / (60 + 4)
r = -48 / 64
r = -3/4
For k = 20, the common ratio is -3/4. The absolute value of -3/4 is 3/4, which is less than 1. Therefore, the geometric sequence converges when k = 20.
Analyzing Convergence for k = 2/3
Now, let's substitute k = 2/3 into the expression for 'r':
r = (12 - 3 * (2/3)) / (3 * (2/3) + 4)
r = (12 - 2) / (2 + 4)
r = 10 / 6
r = 5/3
For k = 2/3, the common ratio is 5/3. The absolute value of 5/3 is 5/3, which is greater than 1. Therefore, the geometric sequence diverges when k = 2/3.
Conclusion on Convergence
Based on our analysis, we conclude that the geometric sequence converges only when k = 20. When k = 2/3, the sequence diverges due to the common ratio having an absolute value greater than 1.
Summary
In this exploration of geometric sequences, we successfully navigated a problem involving the determination of a constant 'k' and the analysis of sequence convergence. We began by establishing the fundamental property of geometric sequences: the constant ratio between consecutive terms. This property led us to derive the quadratic equation 3k² - 62k + 40 = 0, which we then solved using the quadratic formula, obtaining two possible values for 'k': 20 and 2/3.
Subsequently, we delved into the concept of convergence in geometric sequences, highlighting the crucial role of the common ratio 'r'. We established that a geometric sequence converges if and only if |r| < 1. By calculating the common ratio for each value of 'k', we determined that the sequence converges when k = 20 and diverges when k = 2/3.
This problem showcases the interconnectedness of various mathematical concepts, including geometric sequences, quadratic equations, and convergence analysis. By applying fundamental principles and employing algebraic techniques, we were able to unravel the intricacies of the sequence and arrive at meaningful conclusions.
This exploration provides valuable insights into the behavior of geometric sequences and reinforces the importance of understanding key concepts such as the common ratio and convergence criteria. As we continue our mathematical journey, these insights will undoubtedly serve as valuable tools in tackling more complex problems and exploring new mathematical horizons.
