Introduction to Arithmetic Sequences

In the realm of mathematics, arithmetic sequences form a fundamental concept that appears in various practical applications and theoretical studies. Understanding the behaviour of these sequences is crucial for mathematicians, engineers, and students alike. The given problem involves finding the general term of two specific arithmetic sequences: 1/21 and 3/22. This article delves into the process of identifying the common difference and the general term of these sequences, providing a comprehensive understanding of the arithmetic sequence concept.

The General Term of an Arithmetic Sequence

To understand the general term of an arithmetic sequence, it's essential to grasp a few fundamental concepts. An arithmetic sequence is a sequence of numbers in which the difference between any two successive members is a constant. This constant difference is known as the common difference (d). The general term of an arithmetic sequence can be expressed in the form:

[a_n a_1 (n-1)d]

where an represents the nth term of the sequence, a1 is the first term, d is the common difference, and n is the position of the term in the sequence.

Applying the Concept to 1/21 and 3/22

The given problem involves finding the general term of the sequences 1/21 and 3/22. To solve this problem, it's necessary to identify the pattern, determine the common difference, and then write the general term.

Sequence 1: 1/21

Let's identify the common difference in the sequence 1/21. To do this, we need at least two terms of the sequence. Suppose we have the sequence:

[1, 3, 5, 7, 9, ldots]

Here, the first term a1 is 1. Let's calculate the common difference d between the first and second terms:

[d a_2 - a_1 3 - 1 2]

Thus, the common difference d is 2. Using the general term formula, we can write the general term of this arithmetic sequence as:

[a_n 1 (n-1) cdot 2 1 2n - 2 2n - 1]

Sequence 2: 3/22

Similarly, let's consider the sequence 3/22. We assume a pattern and examine the common difference:

[3, 6, 9, 12, 15, ldots]

In this sequence, the first term a1 is 3. The common difference d can be calculated as follows:

[d a_2 - a_1 6 - 3 3]

Using the general term formula, the general term of this arithmetic sequence is:

[a_n 3 (n-1) cdot 3 3 3n - 3 3n]

Conclusion

Arithmetic sequences are a vital aspect of mathematical study, and understanding the general term is crucial for solving many mathematical problems. In this article, we have analyzed the sequences 1/21 and 3/22, identified their common differences, and derived their general terms. These sequences illustrate how the general term formula can be applied to find the nth term of an arithmetic sequence.

For further exploration, students and enthusiasts can experiment with different sequences and common differences to deepen their understanding of arithmetic sequences and their applications.

Keywords: arithmetic sequence, general term, nth term