Finding K Given The Average Of A Sequence K+1 K+2 K+3 K+4 K+5

In the realm of mathematics, particularly in algebra and statistics, the concept of averages plays a pivotal role. Averages, also known as means, provide a measure of central tendency, representing a typical value within a set of numbers. Problems involving averages often require us to manipulate equations and solve for unknown variables, testing our understanding of fundamental mathematical principles. In this article, we will delve into a specific problem concerning the average of a sequence of numbers, where our goal is to determine the value of an unknown variable, K. This problem not only reinforces our understanding of averages but also highlights the importance of algebraic manipulation in problem-solving.

Problem Statement

The problem at hand presents us with a sequence of five consecutive numbers: K+1, K+2, K+3, K+4, and K+5. We are given that the average of these five numbers is equal to 5. Furthermore, we are informed that K is a positive number. Our task is to determine the specific value of K that satisfies these conditions. To solve this problem, we will need to recall the definition of an average and apply algebraic techniques to isolate K.

Understanding the Concept of Average

Before we dive into the solution, let's briefly revisit the concept of average. The average of a set of numbers is calculated by summing all the numbers in the set and then dividing the sum by the total number of elements in the set. In mathematical notation, the average (often denoted as μ) of a set of n numbers (x₁, x₂, ..., xₙ) is given by:

μ = (x₁ + x₂ + ... + xₙ) / n

In our problem, the set of numbers is {K+1, K+2, K+3, K+4, K+5}, and the number of elements in the set is 5. We are given that the average of this set is 5. Therefore, we can set up an equation using the formula for average and solve for K.

Setting up the Equation

Using the definition of average, we can express the average of the given sequence of numbers as follows:

Average = [(K+1) + (K+2) + (K+3) + (K+4) + (K+5)] / 5

We are given that the average is equal to 5. Therefore, we can write the equation:

[(K+1) + (K+2) + (K+3) + (K+4) + (K+5)] / 5 = 5

This equation forms the foundation for solving the problem. Our next step is to simplify the equation by combining like terms and then isolate K to find its value.

Simplifying the Equation

To simplify the equation, we first focus on the numerator, which contains the sum of the five expressions. We can combine the K terms and the constant terms separately:

(K+1) + (K+2) + (K+3) + (K+4) + (K+5) = (K + K + K + K + K) + (1 + 2 + 3 + 4 + 5)

This simplifies to:

5K + 15

Now, we can substitute this simplified expression back into the equation:

(5K + 15) / 5 = 5

To further simplify the equation, we can multiply both sides of the equation by 5 to eliminate the denominator:

5 * [(5K + 15) / 5] = 5 * 5

This gives us:

5K + 15 = 25

Now, we have a simpler linear equation that we can solve for K.

Solving for K

To isolate K, we need to first subtract 15 from both sides of the equation:

5K + 15 - 15 = 25 - 15

This simplifies to:

5K = 10

Finally, to solve for K, we divide both sides of the equation by 5:

5K / 5 = 10 / 5

This gives us the value of K:

K = 2

Therefore, the value of K that satisfies the given conditions is 2. This means that the sequence of numbers is 3, 4, 5, 6, and 7, and their average is indeed 5.

Verifying the Solution

To ensure that our solution is correct, we can substitute K = 2 back into the original equation and check if the average of the sequence is indeed 5. The sequence becomes:

K+1 = 2+1 = 3 K+2 = 2+2 = 4 K+3 = 2+3 = 5 K+4 = 2+4 = 6 K+5 = 2+5 = 7

The sequence is {3, 4, 5, 6, 7}. Now, let's calculate the average:

Average = (3 + 4 + 5 + 6 + 7) / 5

Average = 25 / 5

Average = 5

As we can see, the average of the sequence is indeed 5, which confirms that our solution K = 2 is correct. This verification step is crucial in problem-solving as it helps us identify any potential errors and ensures the accuracy of our answer.

Conclusion

In this article, we tackled a problem involving the average of a sequence of numbers and successfully determined the value of an unknown variable, K. We started by understanding the concept of average and setting up an equation based on the given information. We then simplified the equation using algebraic techniques and solved for K. Finally, we verified our solution to ensure its accuracy. This problem highlights the importance of understanding fundamental mathematical principles, such as averages, and the ability to apply algebraic manipulation to solve problems. The solution process involved several key steps:

1. Understanding the problem statement: We carefully read and interpreted the problem, identifying the given information and the unknown variable we needed to find.
2. Recalling the definition of average: We revisited the concept of average and its formula, which formed the basis for our equation.
3. Setting up the equation: We translated the problem into a mathematical equation using the definition of average and the given information.
4. Simplifying the equation: We combined like terms and used algebraic techniques to simplify the equation and make it easier to solve.
5. Solving for K: We isolated K by performing algebraic operations on both sides of the equation, ultimately finding its value.
6. Verifying the solution: We substituted the value of K back into the original equation to ensure that it satisfied the given conditions.

By following these steps, we were able to confidently solve the problem and determine the value of K. This problem serves as a valuable exercise in applying mathematical principles and problem-solving strategies. Remember, practice and a solid understanding of fundamental concepts are key to success in mathematics.

Options

The correct value of K is 2, which might be presented as one of the options in a multiple-choice question. Other options might include incorrect values obtained through errors in calculation or algebraic manipulation.