Understanding the nth Term of an Arithmetic Sequence: Explained

When dealing with sequences in mathematics, one of the most common questions revolves around determining the nth term. This article aims to clarify the concept of the nth term in the context of the arithmetic sequence given in the sequence 1, 3, 5, 7. We will explore the formula used to find the nth term, its derivation, and provide a few worked examples for better understanding.

Introduction to Arithmetic Sequences

An arithmetic sequence is a list of numbers where the difference between consecutive terms remains constant. This constant difference is known as the common difference, denoted by d. In the sequence 1, 3, 5, 7, the first term a_1 is 1, and the common difference d is 2 (since 3-12, 5-32, and 7-52).

Deriving the nth Term Formula

The formula for finding the nth term of an arithmetic sequence is crucial for many mathematical problems. It is given by:

a_n a_1 (n-1)d

Where:

a_n is the nth term of the sequence. a_1 is the first term of the sequence. n is the term number. d is the common difference.

Applying the Formula to the Given Sequence

Using the given sequence 1, 3, 5, 7, we have:

a_1 1 d 2 Therefore, we can use the formula a_n 1 (n-1)2

Let's simplify this further:

a_n 1 2n - 2 2n - 1

Thus, the nth term of the sequence 1, 3, 5, 7 is given by the formula:

fn 2n - 1

Where fn is the nth term.

Examples and Practice

Let's practice by finding the 5th term of the sequence 1, 3, 5, 7 using our formula:

fn 2n - 1

To find the 5th term:

fn(5) 2(5) - 1 10 - 1 9

The 5th term of the sequence is 9.

Conclusion and Further Exploration

Understanding the nth term of an arithmetic sequence is a fundamental concept in algebra and mathematics. It helps in solving a wide range of problems and is crucial for understanding more complex mathematical topics. By mastering the formula and its application, students can solve various problems efficiently.

Explore further by practicing more problems with different sequences. This will enhance your understanding and speed in solving such questions.

For more resources and detailed explanations, visit reputable educational websites or consult textbooks.