Finding the Term of an Arithmetic Sequence: A Mathematical Exploration

Mathematics, especially in the forms of arithmetic sequences, plays a vital role in understanding patterns, predicting outcomes, and solving real-world problems. In this article, we'll dive into the fascinating world of finding a specific term in an arithmetic sequence. Let's explore a concrete example where we determine the term that equals 401 in the sequence 5, 9, 13, 17, and so on.

Understanding the Sequence: From 5 to 401

Given an arithmetic sequence starting at 5 with a common difference of 4, our goal is to identify the exact term that equals 401. An arithmetic sequence is a list of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the preceding term.

Let's represent the nth term of the sequence as (a_n). The formula for the general term of an arithmetic sequence is given by:

(a_n a_1 (n - 1)d)

where:

a_1 is the first term of the sequence, d is the common difference, and n is the position of the term in the sequence.

Calculating the Position of 401

Given:

a_1 5 d 4 a_n 401

We need to find n such that:

(401 5 (n - 1) cdot 4)

First, let's simplify the equation:

(401 5 4(n - 1))

Next, subtract 5 from both sides:

(396 4(n - 1))

Now, divide both sides by 4:

(99 n - 1)

Add 1 to both sides:

(n 100)

Thus, the 100th term in the sequence is 401. This can also be verified by plugging 100 into the general term formula:

(a_{100} 5 (100 - 1) cdot 4 5 99 cdot 4 5 396 401)

This confirms that 401 is indeed the 100th term in the sequence. Therefore, the answer to the posed question is:

**401 is the 100th term of the arithmetic sequence.**

Conclusion of the Exploration

Through this mathematical exploration, we have gained insight into how to find a specific term in an arithmetic sequence. The method involves utilizing the general term formula for an arithmetic sequence, solving for the unknown variable, and verifying the result. Understanding these concepts is crucial for students and professionals alike who deal with patterns and sequences in various fields, from physics to computer science.

Additional Insights

For those interested in further exploring arithmetic sequences, here are some additional concepts and questions to ponder:

How can you find the sum of the first n terms of an arithmetic sequence? What is the formula to find the nth term of a sequence given the sum of the first n terms? How do you identify whether a sequence is arithmetic or not?

If you found this exploration illuminating, you might also enjoy delving into more advanced topics in sequences and series. Keep learning, exploring, and discovering the joy of mathematics!