Exploring Arithmetic Sequences: Determining the Sequence Given the 4th and 6th Terms

In the realm of mathematics, understanding sequences is fundamental. This article delves into the specifics of an arithmetic sequence, where each term after the first is obtained by adding a constant, known as the common difference, to the previous term. We will explore a particular scenario where the fourth term is 10 and the sixth term is 16. By examining this example, we can determine the entire sequence.

Defining the Problem

Let's consider an arithmetic sequence where the fourth term, denoted as T4, is 10 and the sixth term, denoted as T6, is 16. The general form of the nth term in an arithmetic sequence is given by:

Tn a (n - 1) d

Step-by-Step Solution

Determining the Common Difference (d)

To find the common difference, we can use the given terms. Let's denote the first term as a and the common difference as d. We know:

T4 a (4 - 1)d
10 a 3d

T6 a (6 - 1)d
16 a 5d

Now, let's solve these equations simultaneously. Subtract the equation for the fourth term from the equation for the sixth term:

(16 - 10) (a 5d) - (a 3d)
6 2d
d 3

Solving for the First Term (a)

Now that we know d 3, we can find the first term a. Using the equation for the fourth term:

10 a 3 * 3
10 a 9
a 1

Formulating the Sequence

With the first term a 1 and the common difference d 3, we can now generate the entire sequence. The sequence is:

1, 4, 7, 10, 13, 16, 19, ...

Verification and Visualization

Each term in the sequence is obtained by adding the common difference, which is 3, to the previous term. For example, the fifth term (13) is obtained by adding 3 to the fourth term (10). This pattern continues throughout the sequence.

Understanding the Sequence Properties

Another way to understand this sequence is to note that the difference between consecutive terms is consistently 3. This property is a defining characteristic of an arithmetic sequence. Moreover, the fifth term can be considered to be halfway between the fourth term (10) and the sixth term (16), which is 13. This provides an intuitive way to verify the sequence.

Conclusion

By comprehending the relationship between terms in an arithmetic sequence, we can determine the entire sequence when given specific terms. The sequence where the fourth term is 10 and the sixth term is 16 is easily determined to be 1, 4, 7, 10, 13, 16, 19, ... This article has demonstrated the step-by-step process for finding this sequence, emphasizing the importance of understanding common differences and the general formula for arithmetic sequences.

Related Keywords

arithmetic sequence - a sequence of numbers such that the difference between any two successive members is a constant.

term in sequence - in a sequence, a specific element or number within the set of numbers arranged in a particular order.

common difference - the constant difference between consecutive terms in an arithmetic sequence.