Understanding Arithmetic Sequences: The 4 7 10 13 16 19 Example

The concept of arithmetic sequences is a fundamental part of algebra, often appearing in various mathematical and real-world problems. In this article, we will explore an example of an arithmetic sequence—4, 7, 10, 13, 16, 19—and delve into how to find the common difference, the nth term, and the sum of the sequence.

What is an Arithmetic Sequence?

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers in which the difference between any two successive members is constant. This difference is referred to as the common difference, denoted by (d).

The Example Sequence: 4, 7, 10, 13, 16, 19

Let's take a look at an example sequence: 4, 7, 10, 13, 16, 19. In this sequence, each term is 3 more than the previous term. Therefore, the common difference ((d)) is 3.

Identifying the First Term and the Common Difference

The first term, (a_1): The first term in the sequence is 4. We denote the first term as (a_1 4). The common difference, (d): As mentioned earlier, the common difference in the sequence is 3. We denote the common difference as (d 3).

General Formula for the nth Term

The general formula for the nth term of an arithmetic sequence can be written as:

[a_n a_1 (n - 1)d]

Plugging in the values for our specific example:

[a_n 4 (n - 1) cdot 3]

Expanding the formula:

[a_n 4 3n - 3]

Simplifying the expression:

[a_n 3n 1]

Finding the Next Term

Using the formula, we can easily find the next term in the sequence. For instance, to find the 7th term ((a_7)):

[a_7 3(7) 1 21 1 22]

Thus, the next term in the sequence after 19 is 22.

Sum of the First n Terms

The sum of the first (n) terms of an arithmetic sequence can be calculated using the formula:

[S_n frac{n}{2} [2a_1 (n - 1)d]]

Applying the values from our sequence:

[S_n frac{n}{2} [2(4) (n - 1) cdot 3]]

[S_n frac{n}{2} [8 3n - 3]]

[S_n frac{n}{2} [3n 5]]

[S_n frac{n(3n 5)}{2}]

For instance, if we want to find the sum of the first 6 terms:

[S_6 frac{6(3(6) 5)}{2} 6 cdot frac{23}{2} 3 cdot 23 69]

Further Exploration and Resources

Arithmetic sequences are not only useful in mathematics but also in various applications such as finance, physics, and engineering. If you're interested in further studying these sequences, there are many online resources and tools available. One such resource is the {math website}, which provides detailed explanations and interactive exercises.