Finding The Value Of P In Arithmetic Sequence P+2, 2p-1, P+6

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In mathematics, arithmetic sequences play a fundamental role in understanding patterns and progressions. An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is called the common difference. Understanding arithmetic sequences is crucial not only for grasping basic mathematical concepts but also for solving complex problems in various fields such as physics, engineering, and computer science.

Understanding Arithmetic Sequences

Before diving into the problem, let's solidify our understanding of arithmetic sequences. An arithmetic sequence can be represented as:

  • a, a + d, a + 2d, a + 3d, ...

Where:

  • a is the first term.
  • d is the common difference.

The key characteristic of an arithmetic sequence is that to get from one term to the next, you always add the same value (the common difference). This consistent pattern is what defines the sequence as arithmetic. The common difference, denoted as d, can be found by subtracting any term from its succeeding term. Mathematically, this can be expressed as:

  • d = a₂ - a₁ = a₃ - a₂ = ... = aₙ - aₙ₋₁

Where:

  • a₁, a₂, a₃, ..., aₙ represent the terms of the sequence.

Understanding this fundamental property is crucial for solving problems related to arithmetic sequences. In many problems, you might be given a few terms of a sequence and asked to find the common difference, a specific term, or even a formula for the nth term. Recognizing that the difference between consecutive terms is constant is the first step toward solving these problems.

For instance, consider the sequence 2, 5, 8, 11, .... To verify that this is an arithmetic sequence, we calculate the differences between consecutive terms:

  • 5 - 2 = 3
  • 8 - 5 = 3
  • 11 - 8 = 3

Since the difference is consistently 3, this is indeed an arithmetic sequence with a common difference of 3. This simple example illustrates how we can identify an arithmetic sequence and determine its common difference. The principles and patterns observed in these basic examples will extend to more complex problems, making a solid understanding of arithmetic sequences essential for mathematical proficiency.

Problem Statement: Finding the Value of p

In this article, we will tackle a specific problem involving an arithmetic sequence. The problem states that the first three terms of an arithmetic sequence are given as p+2, 2p-1, and p+6. Our primary goal is to determine the value of p. This problem is a classic example of how we can use the properties of arithmetic sequences to solve for unknown variables. The key to solving this problem lies in understanding and applying the concept of the common difference.

Since the given terms form an arithmetic sequence, the difference between consecutive terms must be constant. This means that the difference between the second term (2p-1) and the first term (p+2) must be equal to the difference between the third term (p+6) and the second term (2p-1). By setting up an equation based on this principle, we can solve for the unknown variable p. This approach leverages the fundamental definition of an arithmetic sequence, which dictates that the difference between successive terms remains constant throughout the sequence. The challenge, then, is to accurately represent these differences in an equation and solve it correctly.

Setting Up the Equation

To find the value of p, we will use the property that the difference between consecutive terms in an arithmetic sequence is constant. This means the difference between the second and first terms should be equal to the difference between the third and second terms. The given terms are:

  • First term: p + 2
  • Second term: 2p - 1
  • Third term: p + 6

We can express the common difference in two ways:

  1. Difference between the second and first terms: (2p - 1) - (p + 2)
  2. Difference between the third and second terms: (p + 6) - (2p - 1)

Since these differences must be equal, we can set up the following equation:

(2p - 1) - (p + 2) = (p + 6) - (2p - 1)

This equation forms the backbone of our solution. It directly applies the defining characteristic of arithmetic sequences—the constant difference between successive terms. By setting these differences equal to each other, we create an algebraic equation that contains the variable p. The subsequent steps will involve simplifying this equation and isolating p to find its value. This equation is a clear representation of the arithmetic sequence property, and its accurate setup is critical for arriving at the correct answer. The forthcoming steps will demonstrate the algebraic manipulations needed to solve this equation, highlighting the methodical approach required in solving mathematical problems.

Solving for p

Now that we have our equation, (2p - 1) - (p + 2) = (p + 6) - (2p - 1), we need to solve for p. This involves simplifying both sides of the equation and isolating p. Let's go through the steps:

  1. Simplify the left side:

    (2p - 1) - (p + 2) = 2p - 1 - p - 2 = p - 3

  2. Simplify the right side:

    (p + 6) - (2p - 1) = p + 6 - 2p + 1 = -p + 7

Now our equation looks like this:

p - 3 = -p + 7

Next, we want to get all the p terms on one side of the equation and the constants on the other side.

  1. Add p to both sides:

    p - 3 + p = -p + 7 + p

    2p - 3 = 7

  2. Add 3 to both sides:

    2p - 3 + 3 = 7 + 3

    2p = 10

Finally, to isolate p, we divide both sides by 2:

  1. Divide both sides by 2:

    2p / 2 = 10 / 2

    p = 5

Thus, the value of p is 5. Each step in this algebraic manipulation is critical to arriving at the correct answer. The process of simplification, combining like terms, and performing inverse operations to isolate the variable is a fundamental skill in algebra. By systematically working through each step, we've successfully found the value of p that satisfies the given conditions of the arithmetic sequence. This methodical approach not only helps in solving this particular problem but also reinforces the broader principles of algebraic problem-solving. The solution provides a clear and concise illustration of how to apply algebraic techniques in the context of arithmetic sequences.

Verification of the Solution

To ensure the correctness of our solution, it's crucial to verify whether the value of p we found indeed results in an arithmetic sequence. We found that p = 5. Let's substitute this value back into the original terms of the sequence:

  • First term: p + 2 = 5 + 2 = 7
  • Second term: 2p - 1 = 2(5) - 1 = 10 - 1 = 9
  • Third term: p + 6 = 5 + 6 = 11

Now we have the sequence 7, 9, 11. To check if this is an arithmetic sequence, we need to see if the difference between consecutive terms is constant:

  • Difference between the second and first terms: 9 - 7 = 2
  • Difference between the third and second terms: 11 - 9 = 2

Since the difference is consistently 2, the sequence 7, 9, 11 is an arithmetic sequence. This confirms that our value of p = 5 is correct. The process of verification is a critical step in problem-solving, as it provides a means of checking for errors and ensuring the accuracy of the final answer. By substituting the solved value back into the original problem, we can confirm whether the conditions of the problem are satisfied. In this case, verifying that the resulting sequence is indeed arithmetic validates our solution and demonstrates a thorough approach to mathematical problem-solving.

Conclusion

In this article, we solved a problem involving an arithmetic sequence by determining the value of p given the first three terms: p+2, 2p-1, and p+6. We utilized the fundamental property of arithmetic sequences—the constant difference between consecutive terms—to set up an equation and solve for p. Our step-by-step solution involved simplifying the equation, isolating the variable, and arriving at the value p = 5. We then verified our solution by substituting p back into the original terms and confirming that the resulting sequence (7, 9, 11) is indeed arithmetic.

This problem highlights the importance of understanding the properties of arithmetic sequences and applying algebraic techniques to solve mathematical problems. The ability to recognize patterns, set up equations, and systematically solve them is a crucial skill in mathematics and various other fields. By verifying our solution, we reinforced the accuracy of our answer and demonstrated a comprehensive approach to problem-solving. Mastering these concepts and skills can greatly enhance mathematical proficiency and the ability to tackle more complex problems in the future. The solution presented here serves as a clear example of how to approach problems involving arithmetic sequences, providing a solid foundation for further exploration of mathematical concepts.