The Explicit Formula for an Arithmetic Sequence

Oct 13, 2025 by leolavanderia.com.br 8 views

Introduction

Arithmetic sequences are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. 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 explicit formula for the arithmetic sequence $$5, 10, 15, 20, \ldots$$ and examine the different options provided.

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, in the sequence $$5, 10, 15, 20, \ldots$$, the common difference is $$5$$, as each term is obtained by adding $$5$$ to the previous term.

The Explicit Formula for an Arithmetic Sequence

The explicit 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.

Deriving the Explicit Formula for the Given Sequence

To derive the explicit formula for the given sequence $$5, 10, 15, 20, \ldots$$, we need to identify the first term and the common difference. The first term is $$5$$, and the common difference is $$5$$.

Using the explicit formula, we can write:

a_n = 5 + (n - 1)5

Simplifying the expression, we get:

a_n = 5 + 5n - 5

a_n = 5n

Evaluating the Options

Now that we have derived the explicit formula for the given sequence, let's evaluate the options provided:

A. $$a_n = 5n$$

B. $$a_n = 5n + 1$$

C. $$a_n = 5n - 5$$

D. $$a_n = 5n - 1$$

Based on our derivation, we can see that option A is the correct explicit formula for the given sequence.

Conclusion

In conclusion, the explicit formula for the arithmetic sequence $$5, 10, 15, 20, \ldots$$ is $$a_n = 5n$$. This formula can be derived using the explicit formula for an arithmetic sequence, which is given by $$a_n = a_1 + (n - 1)d$$. We hope that this article has provided a clear understanding of arithmetic sequences and the explicit formula for the given sequence.

Frequently Asked Questions

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.

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

A: The common difference is the constant difference between any two consecutive terms in an arithmetic sequence.

Q: How do I derive the explicit formula for an arithmetic sequence?

A: To derive the explicit formula for an arithmetic sequence, you need to identify the first term and the common difference, and then use the explicit formula $$a_n = a_1 + (n - 1)d$$.

Q: What is the explicit formula for the given sequence $$5, 10, 15, 20, \ldots$$?

A: The explicit formula for the given sequence is $$a_n = 5n$$.

References

[1] "Arithmetic Sequences and Series" by Math Open Reference
[2] "Arithmetic Sequences" by Purplemath
[3] "Arithmetic Sequences and Series" by Khan Academy
Arithmetic Sequences Q&A: Frequently Asked Questions =====================================================

Introduction

Arithmetic sequences are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. In our previous article, we explored the explicit formula for the arithmetic sequence $$5, 10, 15, 20, \ldots$$ and examined the different options provided. In this article, we will answer some frequently asked questions about arithmetic sequences.

Q&A

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.

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

A: The common difference is the constant difference between any two consecutive terms in an arithmetic sequence.

Q: How do I determine if a sequence is an arithmetic sequence?

A: To determine if a sequence is an arithmetic sequence, you need to check if the difference between any two consecutive terms is constant. If it is, then the sequence is an arithmetic sequence.

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

A: To find the nth term of an arithmetic sequence, you can use the explicit formula $$a_n = a_1 + (n - 1)d$$, where $$a_n$$ is the nth term, $$a_1$$ is the first term, $$n$$ is the term number, and $$d$$ is the common difference.

Q: What is the formula for the sum of the first n terms of an arithmetic sequence?

A: The formula for the sum of the first n terms of an arithmetic sequence is $$S_n = \frac{n}{2}(a_1 + a_n)$$, where $$S_n$$ is the sum of the first n terms, $$a_1$$ is the first term, $$a_n$$ is the nth term, and $$n$$ is the term number.

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

A: To find the sum of an infinite arithmetic sequence, you need to use the formula $$S = \frac{a_1 + a_2}{2}$$, where $$S$$ is the sum, $$a_1$$ is the first term, and $$a_2$$ is the second term.

Q: What is the formula for the nth term of an arithmetic sequence in terms of the sum of the first n terms?

A: The formula for the nth term of an arithmetic sequence in terms of the sum of the first n terms is $$a_n = S_n - S_{n-1}$$, where $$a_n$$ is the nth term, $$S_n$$ is the sum of the first n terms, and $$S_{n-1}$$ is the sum of the first n-1 terms.

Q: How do I find the sum of the first n terms of an arithmetic sequence using the formula $$S_n = \frac{n}{2}(a_1 + a_n)$$?

A: To find the sum of the first n terms of an arithmetic sequence using the formula $$S_n = \frac{n}{2}(a_1 + a_n)$$, you need to substitute the values of $$a_1$$, $$a_n$$, and $$n$$ into the formula and simplify.

Q: What is the formula for the sum of the first n terms of an arithmetic sequence in terms of the first term and the common difference?

A: The formula for the sum of the first n terms of an arithmetic sequence in terms of the first term and the common difference is $$S_n = \frac{n}{2}[2a_1 + (n - 1)d]$$, where $$S_n$$ is the sum of the first n terms, $$a_1$$ is the first term, $$d$$ is the common difference, and $$n$$ is the term number.

Conclusion

In conclusion, arithmetic sequences are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. We hope that this article has provided a clear understanding of arithmetic sequences and answered some frequently asked questions.

Frequently Asked Questions (FAQs)

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.

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

A: The common difference is the constant difference between any two consecutive terms in an arithmetic sequence.

Q: How do I determine if a sequence is an arithmetic sequence?

A: To determine if a sequence is an arithmetic sequence, you need to check if the difference between any two consecutive terms is constant. If it is, then the sequence is an arithmetic sequence.

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

References

[1] "Arithmetic Sequences and Series" by Math Open Reference
[2] "Arithmetic Sequences" by Purplemath
[3] "Arithmetic Sequences and Series" by Khan Academy