Solving for k in an Arithmetic Sequence

Introduction

Arithmetic sequences are an important concept in mathematics, where the difference between consecutive terms is constant. In this article, we explore the problem of finding the value of k such that the terms (5k - 3), (k 5), and (2k - 2) form an arithmetic sequence. We will walk through the steps to solve this problem and demonstrate the application of arithmetic sequence properties and linear equations.

Understanding the Problem

To ensure that (5k - 3), (k 5), and (2k - 2) form an arithmetic sequence, we need to use the property that the difference between consecutive terms is constant. Mathematically, this means:

[k 5 - (5k - 3) (2k - 2) - (k 5)]

This equation represents the condition that the difference between the first and second term is equal to the difference between the second and third term in the sequence.

Simplifying the Equation

Let's simplify both sides of the equation:

Left side:

[k 5 - (5k - 3) k 5 - 5k 3 -4k 8]

Right side:

[(2k - 2) - (k 5) 2k - 2 - k - 5 k - 7]

Now, set the two sides equal to each other:

[-4k 8 k - 7]

Solving for k

To find the value of k, we solve the linear equation:

Add (4k) to both sides:

[8 5k - 7]

Add (7) to both sides:

[15 5k]

Divide by (5):

[k 3]

Thus, the value of k that makes the terms (5k - 3), (k 5), and (2k - 2) form an arithmetic sequence is boxed{3}.

Verification

Let's verify this result:

For (k 3), the terms become:

[5k - 3 15 - 3 12]

[k 5 3 5 8]

[2k - 2 2(3) - 2 6 - 2 4]

The terms (4), (8), and (12) form an arithmetic sequence with a common difference of 4.

Conclusion

By solving the linear equation and verifying the result, we demonstrated that (k 3) is the correct value to make the terms (5k - 3), (k 5), and (2k - 2) form an arithmetic sequence. This problem helps us understand the application of arithmetic sequence properties and the solution of linear equations.