Finding K For Consecutive AP Terms | Arithmetic Progression Problem
In this article, we delve into the fascinating world of arithmetic progressions (APs) and explore how to determine the value of k such that the expressions k^2 + 4k + 8, 2k^2 + 3k + 6, and 3k^2 + 4k + 4 form three consecutive terms of an AP. This problem elegantly combines algebraic manipulation with the fundamental properties of arithmetic sequences. Our journey will involve understanding the core concept of APs, setting up equations based on the given terms, and employing algebraic techniques to solve for k. By the end of this exploration, you'll have a solid grasp of how to tackle similar problems involving arithmetic progressions and quadratic expressions.
Understanding Arithmetic Progressions
Before we dive into the specifics of the problem, let's first solidify our understanding of arithmetic progressions. An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference, often denoted by d. For example, the sequence 2, 5, 8, 11, 14 is an AP with a common difference of 3.
The general form of an AP can be written as:
- a, a + d, a + 2d, a + 3d, ...
where a is the first term and d is the common difference.
A crucial property of arithmetic progressions is that the difference between any two consecutive terms is the same. Mathematically, if we have three consecutive terms x, y, and z in an AP, then:
- y - x = z - y
This relationship forms the cornerstone of our approach to solving the problem at hand. We will use this property to set up an equation involving the given expressions and then solve for the unknown variable, k. Recognizing this fundamental characteristic of APs is essential for tackling problems where you need to determine if a given sequence is arithmetic or find missing terms in an arithmetic sequence.
Now, let's connect this understanding to the given problem. We are presented with three expressions involving k, and we need to find the value(s) of k that make these expressions consecutive terms of an AP. This means that the difference between the second and first expressions must be equal to the difference between the third and second expressions. This sets the stage for the algebraic manipulation we will perform in the subsequent sections.
Setting up the Equation
Now that we have a solid understanding of arithmetic progressions, let's apply this knowledge to the problem at hand. We are given three expressions:
- k^2 + 4k + 8
- 2k^2 + 3k + 6
- 3k^2 + 4k + 4
and we need to find the value(s) of k that make these expressions consecutive terms of an arithmetic progression. As we discussed earlier, the key property of an AP is that the difference between consecutive terms is constant. Therefore, we can set up the following equation:
(2k^2 + 3k + 6) - (k^2 + 4k + 8) = (3k^2 + 4k + 4) - (2k^2 + 3k + 6)
This equation represents the core relationship that must hold true if the given expressions are indeed consecutive terms of an AP. The left-hand side of the equation represents the difference between the second and first terms, while the right-hand side represents the difference between the third and second terms. If these differences are equal, it confirms that the terms form an AP.
Now, our task is to simplify this equation and solve for k. This will involve carefully expanding the expressions, combining like terms, and potentially using factoring or the quadratic formula to find the solution(s) for k. The process of simplifying and solving the equation is a crucial step in determining the value(s) of k that satisfy the given condition. By setting up the equation correctly and employing accurate algebraic techniques, we can arrive at the solution and identify the value(s) of k that make the expressions consecutive terms in an arithmetic progression.
Solving for k
With the equation set up, the next step is to solve for k. Let's begin by simplifying the equation:
(2k^2 + 3k + 6) - (k^2 + 4k + 8) = (3k^2 + 4k + 4) - (2k^2 + 3k + 6)
First, we'll remove the parentheses, being mindful of the negative signs:
2k^2 + 3k + 6 - k^2 - 4k - 8 = 3k^2 + 4k + 4 - 2k^2 - 3k - 6
Next, we combine like terms on both sides of the equation:
(2k^2 - k^2) + (3k - 4k) + (6 - 8) = (3k^2 - 2k^2) + (4k - 3k) + (4 - 6)
This simplifies to:
k^2 - k - 2 = k^2 + k - 2
Now, we can subtract k^2 from both sides:
-k - 2 = k - 2
Next, we add k to both sides:
-2 = 2k - 2
Then, we add 2 to both sides:
0 = 2k
Finally, we divide both sides by 2:
k = 0
Therefore, the value of k that makes the given expressions consecutive terms of an arithmetic progression is k = 0. It's crucial to verify this solution by substituting k = 0 back into the original expressions to ensure they indeed form an AP.
Verifying the Solution
After solving for k, it's crucial to verify our solution. This involves substituting the value we found (k = 0) back into the original expressions and checking if they form an arithmetic progression. The original expressions are:
- k^2 + 4k + 8
- 2k^2 + 3k + 6
- 3k^2 + 4k + 4
Substituting k = 0 into these expressions, we get:
- (0)^2 + 4(0) + 8 = 8
- 2(0)^2 + 3(0) + 6 = 6
- 3(0)^2 + 4(0) + 4 = 4
So, the terms become 8, 6, and 4. Now, we need to check if these terms form an AP. To do this, we examine the differences between consecutive terms:
- 6 - 8 = -2
- 4 - 6 = -2
Since the differences between consecutive terms are equal (-2), the terms 8, 6, and 4 do indeed form an arithmetic progression. This confirms that our solution k = 0 is correct.
Verifying the solution is an essential step in problem-solving, especially in mathematics. It helps to catch any potential errors made during the algebraic manipulation or equation-solving process. By substituting the solution back into the original problem and checking if it satisfies the given conditions, we can be confident in the accuracy of our answer. In this case, the verification step provides assurance that k = 0 is the correct value that makes the given expressions consecutive terms of an AP.
Conclusion
In this article, we successfully determined the value of k that makes the expressions k^2 + 4k + 8, 2k^2 + 3k + 6, and 3k^2 + 4k + 4 consecutive terms of an arithmetic progression. Our journey involved understanding the fundamental properties of arithmetic progressions, setting up an equation based on the constant difference between consecutive terms, solving the equation using algebraic techniques, and verifying the solution by substituting it back into the original expressions.
The key steps we undertook were:
- Understanding Arithmetic Progressions: We revisited the definition of an AP and the property that the difference between consecutive terms is constant.
- Setting up the Equation: We used the property of constant difference to create an equation relating the given expressions.
- Solving for k: We simplified the equation and solved for k using algebraic manipulation.
- Verifying the Solution: We substituted the value of k back into the original expressions to ensure they formed an AP.
The result we obtained was k = 0. This value, when substituted into the expressions, yields the arithmetic progression 8, 6, and 4, with a common difference of -2.
This problem exemplifies how algebraic techniques can be applied to solve problems involving arithmetic sequences. The ability to translate the properties of mathematical sequences into algebraic equations is a valuable skill in problem-solving. Furthermore, the importance of verifying the solution cannot be overstated, as it ensures the accuracy and validity of the result. By mastering these concepts and techniques, you can confidently tackle a wide range of problems involving arithmetic progressions and other mathematical sequences.
