The General Formula for an Arithmetic Sequence

Oct 13, 2025 by miodeya.com 52 views

Introduction

In mathematics, sequences are an essential concept that helps us understand patterns and relationships between numbers. A sequence is a list of numbers in a specific order, and each number in the list is called a term. In this article, we will analyze a given sequence and determine which function represents it.

The Given Sequence

The given sequence is: $${-1, 2, 5, 8, 11, \ldots}$$

This sequence appears to be a simple arithmetic sequence, where each term is obtained by adding a fixed constant to the previous term.

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In other words, if we take any two consecutive terms, say $a_n$ and $a_{n+1}$ , the difference between them is always the same.

The General Formula for an Arithmetic Sequence

The general formula for an arithmetic sequence is given by:

$a_n = a_1 + (n-1)d$

where $a_n$ is the $n^{th}$ term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Analyzing the Given Sequence

Let's analyze the given sequence and try to find the common difference.

Term Number	Term Value
1	-1
2	2
3	5
4	8
5	11

From the table above, we can see that the difference between consecutive terms is:

$2 - (-1) = 3$ $5 - 2 = 3$ $8 - 5 = 3$ $11 - 8 = 3$

The common difference is 3.

Finding the Function that Represents the Sequence

Now that we have found the common difference, we can use the general formula for an arithmetic sequence to find the function that represents the sequence.

$a_n = a_1 + (n-1)d$

We know that $a_1 = -1$ and $d = 3$ . Plugging these values into the formula, we get:

$a_n = -1 + (n-1)3$

Simplifying the equation, we get:

$a_n = -1 + 3n - 3$

$a_n = 3n - 4$

Therefore, the function that represents the sequence is:

$a_n = 3n - 4$

Conclusion

In this article, we analyzed a given sequence and determined which function represents it. We found that the sequence is an arithmetic sequence with a common difference of 3. Using the general formula for an arithmetic sequence, we found that the function that represents the sequence is $a_n = 3n - 4$ .

Discussion

The sequence we analyzed is a simple arithmetic sequence, but it can be used to model real-world situations. For example, if we have a list of numbers that represent the heights of people in a population, we can use an arithmetic sequence to model the distribution of heights.

References

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

Additional Resources

[1] "Arithmetic Sequences and Series" by Mathway
[2] "Sequences and Series" by Wolfram Alpha

Final Thoughts

In conclusion, understanding sequences and series is an essential part of mathematics. By analyzing a given sequence, we can determine which function represents it and use it to model real-world situations.