Introduction to Arithmetic Sequences and Series

Arithmetic sequences and series are fundamental concepts in mathematics with wide applications in various fields. An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant to the previous term. This constant is known as the common difference. The general formula to find the nth term of an arithmetic sequence is given by:

an a1 (n-1)d

where an is the nth term, a1 is the first term, and d is the common difference.

Finding the First Term of an Arithmetic Sequence

Let's walk through a specific example to illustrate the process of finding the first term of an arithmetic sequence when given the 8th term and the common difference:

Problem Statement: The 8th term of an arithmetic sequence is 23, and the common difference is 3. Using the Formula: To find the first term, we can use the general formula for the nth term of an arithmetic sequence:

23 a1 (8-1)*3

Let's simplify and solve for the first term:

23 a1 7*3

23 a1 21

a1 23 - 21

a1 2

Thus, the first term is 2.

Arithmetic Sequence Example Walkthrough

To further clarify, let's look at the arithmetic sequence step-by-step:

The first term is 2. The common difference is 3. The sequence continues as follows: 2, 5, 8, 11, 14, 17, 20, 23, .... The general term rule can be given as:

tn 2 (n-1)*3

or alternatively, using the common form:

tn 3n - 1

Conclusion and Further Reading

Understanding and mastering arithmetic sequences is crucial for solving a variety of mathematical problems. Whether you're dealing with simple sequences or more complex series, the principles outlined here provide a solid foundation.

For further exploration of related topics, consider checking out the following:

Understanding Arithmetic Sequences Arithmetic Series Common Difference in Arithmetic Sequences

Note: The links provided above are placeholders and should be replaced with actual, relevant resources.