Solving the Common Difference in an Arithmetic Sequence

Arithmetic sequences are essential in various mathematical problems and real-world applications. In this article, we will delve into how to solve for the common difference of an arithmetic sequence given the first three terms: 3k^2-1, 7, and k^2. By understanding this concept, we will not only find the common difference but also explore the algebraic methods necessary to solve such problems.

Understanding Arithmetic Sequences

Before diving into the solution, it is crucial to understand the fundamental concept of an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant, known as the common difference, d. This property can be expressed mathematically as:

[a_n a_{n-1} d]

Here, a_n and a_{n-1} represent the nth and the (n-1)th terms of the sequence, respectively. This constant difference is the basis for solving arithmetic problems involving terms.

Determining the Terms of the Sequence

The first three terms of the given arithmetic sequence are: 3k^2-1, 7, and k^2. We need to find the common difference, d, using these terms. Let's denote the terms as follows:

a_1 3k^2 - 1 a_2 7 a_3 k^2

Using the Common Difference Formula

The common difference, d, can be calculated using consecutive terms as follows:

d a_2 - a_1 d a_3 - a_2

Since these expressions must be equal, we set them equal to each other:

[a_2 - a_1 a_3 - a_2]

Solving this equation will help us determine the value of k^2, and subsequently, the common difference, d.

Setting Up the Equation and Solving for k

Substitute the known terms into the equation:

[7 - (3k^2 - 1) k^2 - 7]

First, simplify the left-side of the equation:

[7 - 3k^2 1 k^2 - 7] [8 - 3k^2 k^2 - 7]

Next, combine like terms:

[8 7 k^2 3k^2] [15 4k^2]

Divide both sides by 4 to isolate k^2:

[k^2 frac{15}{4}]

Calculate the value of k^2:

[k^2 3.75]

Finding the Actual Terms of the Sequence

With the value of k^2 determined, we can now find the actual terms of the sequence:

a_1 3(3.75) - 1 11.25 - 1 10.25 a_2 7 a_3 3.75

The first three terms of the sequence are 10.25, 7, and 3.75, respectively.

Verifying the Common Difference

To ensure the solution is correct, we can calculate the common difference using consecutive terms:

d a_2 - a_1 7 - 10.25 -3.25 d a_3 - a_2 3.75 - 7 -3.25

Both calculations confirm that the common difference, d, is -3.25.

Conclusion

This problem demonstrates the use of algebraic techniques to find the common difference of an arithmetic sequence given the first three terms. By solving for k^2, we were able to determine the actual terms and verify the common difference. Understanding these methods can help in various mathematical and real-world applications involving arithmetic sequences.