Understanding the nth Term of an Arithmetic Sequence: A Comprehensive Guide

The nth term of an arithmetic sequence refers to a formula that allows you to find any term within the sequence without having to know the terms that come before it. This article aims to provide a detailed explanation along with examples to help you understand and apply the concept effectively.

Introduction to Arithmetic Sequences

An arithmetic sequence is defined as a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the preceding term. For instance, in the sequence 7, 9, 11, 13, 15, the common difference is 2.

Identifying the First Term and the Common Difference

Let's start by identifying the first term and the common difference in the sequence given:

First term (a): The first term is 7. Common difference (d): The difference between consecutive terms is 9 - 7 2.

Once we have the first term and the common difference, we can use the formula for the nth term of an arithmetic sequence:

Formula for the nth Term of an Arithmetic Sequence

an a (n - 1)d

This formula can be broken down as follows:

a: The first term of the sequence. d: The common difference between the terms. n: The term number you are trying to find.

Substituting the values of (a) and (d) into the formula:

[(a_n 7 (n - 1) cdot 2)]

Simplifying this expression:

[(a_n 7 2n - 2 2n 5)]

Therefore, the nth term of the arithmetic sequence is:

[boxed{2n 5}]

Verification and Example

To verify that the formula is correct, we can substitute different values of (n). For example:

If (n 1), then (a_1 2 cdot 1 5 7) If (n 2), then (a_2 2 cdot 2 5 9) If (n 3), then (a_3 2 cdot 3 5 11)

As we can see, these values match the terms in the sequence 7, 9, 11, etc.

Further Examples

Another example to illustrate the concept is the sequence 8, 11, 14, 17:

The first term (a 8) The common difference (d 11 - 8 3)

Using the nth term formula:

[(a_n 8 (n - 1) cdot 3 8 3n - 3 3n 5)]

So, the nth term is (3n 5).

Conclusion

Understanding the nth term of an arithmetic sequence is crucial for solving various mathematical problems and for advancing in more complex algebraic concepts. Whether you're a student, a teacher, or a professional needing to work with sequences, knowing how to find the nth term can save time and enhance your problem-solving skills.