Not identifying the common difference correctly

Oct 13, 2025 by windailys.com 69 views

What is the Explicit Formula for the Arithmetic Sequence in the Table Below?

Understanding Arithmetic Sequences

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. Arithmetic sequences are an essential concept in mathematics, and they have numerous applications in various fields, including finance, physics, and engineering.

Identifying the Common Difference

To find the explicit formula for an arithmetic sequence, we need to identify the common difference. The common difference can be found by subtracting any term from its previous term. In the given table, we can see that the difference between the first and second term is 1.8, the difference between the second and third term is also 1.8, and so on.

Finding the Explicit Formula

The explicit formula for an arithmetic sequence is given by the formula:

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.

Applying the Formula to the Given Table

In the given table, the first term $a_1$ is 9.2, and the common difference $d$ is 1.8. We can now apply the formula to find the explicit formula for the arithmetic sequence.

a_n = 9.2 + (n - 1)(1.8)

Simplifying the formula, we get:

a_n = 9.2 + 1.8n - 1.8

a_n = 7.4 + 1.8n

Comparing with the Given Option

The given option is:

a_n = 1 + 1.8(n - 1)

Simplifying the given option, we get:

a_n = 1 + 1.8n - 1.8

a_n = -0.8 + 1.8n

Comparing the two formulas, we can see that they are not the same. However, we can rewrite the given option to match the formula we derived:

a_n = 1 + 1.8(n - 1)

a_n = 1 + 1.8n - 1.8

a_n = -0.8 + 1.8n

Conclusion

In conclusion, the explicit formula for the arithmetic sequence in the given table is:

a_n = 7.4 + 1.8n

This formula can be used to find any term of the sequence by substituting the value of $n$ into the formula.

Discussion

The explicit formula for an arithmetic sequence is a powerful tool that can be used to solve a wide range of problems. In this article, we derived the explicit formula for the arithmetic sequence in the given table and compared it with the given option. We also discussed the importance of arithmetic sequences and their applications in various fields.

Frequently Asked Questions

What is an arithmetic sequence? An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
How do I find the common difference of an arithmetic sequence? The common difference can be found by subtracting any term from its previous term.
What is the explicit formula for an arithmetic sequence? The explicit formula for an arithmetic sequence is given by the formula: $a_n = a_1 + (n - 1)d$ , 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.

References

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

Frequently Asked Questions

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.
How do I find the common difference of an arithmetic sequence? The common difference can be found by subtracting any term from its previous term. For example, if the first term is 5 and the second term is 7, the common difference is 2.
What is the explicit formula for an arithmetic sequence? The explicit formula for an arithmetic sequence is given by the formula: $a_n = a_1 + (n - 1)d$ , 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.
How do I find the first term of an arithmetic sequence? The first term of an arithmetic sequence can be found by looking at the first term in the sequence. For example, if the sequence is 2, 5, 8, 11, 14, the first term is 2.
How do I find the common difference of an arithmetic sequence? The common difference can be found by subtracting any term from its previous term. For example, if the first term is 5 and the second term is 7, the common difference is 2.
What is the formula for the nth term of an arithmetic sequence? The formula for the nth term of an arithmetic sequence is given by the formula: $a_n = a_1 + (n - 1)d$ , 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.
How do I find the sum of an arithmetic sequence? The sum of an arithmetic sequence can be found by using the formula: $S_n = \frac{n}{2}(a_1 + a_n)$ , 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 number of terms.
What is the formula for the sum of an arithmetic sequence? The formula for the sum of an arithmetic sequence is given by the formula: $S_n = \frac{n}{2}(a_1 + a_n)$ , 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 number of terms.
How do I find the number of terms in an arithmetic sequence? The number of terms in an arithmetic sequence can be found by using the formula: $n = \frac{a_n - a_1}{d} + 1$ , where $n$ is the number of terms, $a_n$ is the nth term, $a_1$ is the first term, and $d$ is the common difference.

Common Mistakes to Avoid

Not identifying the common difference correctly: Make sure to subtract the correct terms to find the common difference.
Not using the correct formula: Use the correct formula for the nth term and the sum of an arithmetic sequence.
Not plugging in the correct values: Make sure to plug in the correct values for the first term, common difference, and number of terms.

Tips and Tricks

Use a calculator to check your work: Use a calculator to check your work and make sure you are getting the correct answers.
Check your units: Make sure to check your units and make sure they are correct.
Use a formula chart: Use a formula chart to help you remember the formulas for the nth term and the sum of an arithmetic sequence.

Real-World Applications

Finance: Arithmetic sequences are used in finance to calculate interest rates and investments.
Physics: Arithmetic sequences are used in physics to calculate the motion of objects.
Engineering: Arithmetic sequences are used in engineering to calculate the stress and strain on materials.

Conclusion

Arithmetic sequences are a fundamental concept in mathematics, and they have numerous real-world applications. By understanding the formulas and concepts, you can solve a wide range of problems and make informed decisions. Remember to avoid common mistakes, use a calculator to check your work, and use a formula chart to help you remember the formulas.