Formula for an Arithmetic Sequence

Oct 13, 2025 by windailys.com 72 views

Introduction

Arithmetic sequences are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In this article, we will explore the concept of arithmetic sequences and how to find the fifth term of a given sequence.

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. For example, consider the sequence 2, 5, 8, 11, 14, ... . In this sequence, the common difference is 3, which is the difference between any two consecutive terms.

Formula for an Arithmetic Sequence

The formula for an arithmetic sequence 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, $n$ is the term number, and $d$ is the common difference.

Finding the Fifth Term of an Arithmetic Sequence

To find the fifth term of an arithmetic sequence, we need to substitute $n=5$ into the formula for the nth term. Let's consider the given sequence $a_n = 6n - 4$ . We need to find the fifth term of this sequence.

Step 1: Substitute n=5 into the formula

Substituting $n=5$ into the formula $a_n = 6n - 4$ , we get:

a_5 = 6(5) - 4

Step 2: Simplify the expression

Simplifying the expression, we get:

a_5 = 30 - 4

Step 3: Evaluate the expression

Evaluating the expression, we get:

a_5 = 26

Conclusion

In this article, we have explored the concept of arithmetic sequences and how to find the fifth term of a given sequence. We have used the formula for an arithmetic sequence to find the fifth term of the sequence $a_n = 6n - 4$ . The fifth term of this sequence is 26.

Final Answer

The final answer is 26.

Discussion

The discussion category for this article is mathematics. The article is relevant to students and professionals who are interested in mathematics, particularly arithmetic sequences.

References

[1] "Arithmetic Sequences and Series" by Paul Dawkins
[2] "Mathematics for Dummies" by Mary Jane Sterling
[3] "Calculus" by Michael Spivak

Keywords

Arithmetic sequence
Fifth term
Formula
Common difference
Term number
Sequence
Mathematics

Introduction

Arithmetic sequences are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will answer some frequently asked questions about arithmetic sequences.

Q: What is an arithmetic sequence?

A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.

Q: What is the formula for an arithmetic sequence?

A: The formula for an arithmetic sequence 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, $n$ is the term number, and $d$ is the common difference.

Q: How do I find the nth term of an arithmetic sequence?

A: To find the nth term of an arithmetic sequence, you need to substitute $n$ into the formula for the nth term. For example, if you want to find the 5th term of the sequence $a_n = 6n - 4$ , you would substitute $n=5$ into the formula.

Q: What is the common difference in an arithmetic sequence?

A: The common difference in an arithmetic sequence is the difference between any two consecutive terms. For example, in the sequence 2, 5, 8, 11, 14, ... , the common difference is 3.

Q: How do I find the sum of an arithmetic sequence?

A: To find the sum of an arithmetic sequence, you need to use the formula for the sum of an arithmetic series:

S_n = \frac{n}{2}(a_1 + a_n)

where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the nth term.

Q: What is the difference between an arithmetic sequence and a geometric sequence?

A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant, while a geometric sequence is a sequence of numbers in which the ratio between any two consecutive terms is constant.

Q: Can I use the formula for an arithmetic sequence to find the sum of a geometric sequence?

A: No, you cannot use the formula for an arithmetic sequence to find the sum of a geometric sequence. The formula for an arithmetic sequence is only applicable to arithmetic sequences, while the formula for a geometric sequence is only applicable to geometric sequences.

Q: How do I find the nth term of a geometric sequence?

A: To find the nth term of a geometric sequence, you need to use the formula for the nth term of a geometric sequence:

a_n = a_1 \times r^{n-1}

where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: What is the common ratio in a geometric sequence?

A: The common ratio in a geometric sequence is the ratio between any two consecutive terms. For example, in the sequence 2, 6, 18, 54, 162, ... , the common ratio is 3.

Q: How do I find the sum of a geometric sequence?

A: To find the sum of a geometric sequence, you need to use the formula for the sum of a geometric series:

S_n = \frac{a_1(1-r^n)}{1-r}

where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Conclusion

In this article, we have answered some frequently asked questions about arithmetic sequences. We have covered topics such as the formula for an arithmetic sequence, finding the nth term, the common difference, and the sum of an arithmetic sequence. We have also compared arithmetic sequences with geometric sequences and provided formulas for finding the nth term and the sum of a geometric sequence.