Considering The Arithmetic Sequence: \[$11 \frac{1}{2}, 9 \frac{5}{16}, 7 \frac{1}{8}, \ldots\$\]What Is The Common Difference Between Each Term?A. \[$-2 \frac{3}{16}\$\]B. 2C. \[$\frac{3}{16}\$\]D. \[$-\frac{3}{16}\$\]
Introduction
Arithmetic sequences are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, geometry, and other branches of mathematics. 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 focus on finding the common difference between each term in 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.
Finding the Common Difference
To find the common difference in an arithmetic sequence, we can use the following formula:
d = a_n - a_(n-1)
where d is the common difference, a_n is the nth term, and a_(n-1) is the (n-1)th term.
Example: Finding the Common Difference
Let's consider the given sequence: 11 1/2, 9 5/16, 7 1/8, ... . To find the common difference, we can use the formula above.
First, we need to convert the mixed numbers to improper fractions. The sequence becomes:
22/2, 149/16, 57/8, ...
Now, we can find the common difference by subtracting each term from the previous term.
d = (149/16) - (22/2) d = (149/16) - (88/16) d = (61/16)
So, the common difference is 61/16.
Simplifying the Common Difference
To simplify the common difference, we can convert the improper fraction to a mixed number.
d = 61/16 d = 3 13/16
Therefore, the common difference is 3 13/16.
Conclusion
In this article, we explored the concept of arithmetic sequences and focused on finding the common difference between each term in a given sequence. We used the formula d = a_n - a_(n-1) to find the common difference and simplified the result to a mixed number. The common difference is a crucial concept in arithmetic sequences, and understanding it is essential for solving various problems in mathematics.
Answer
The correct answer is C. 3 13/16.
Discussion
The common difference is a fundamental concept in arithmetic sequences, and understanding it is essential for solving various problems in mathematics. In this article, we explored the concept of arithmetic sequences and focused on finding the common difference between each term in a given sequence. We used the formula d = a_n - a_(n-1) to find the common difference and simplified the result to a mixed number.
Common Differences in Real-World Applications
Arithmetic sequences and their common differences have numerous real-world applications. For example, in finance, the common difference can be used to calculate the interest rate on a loan or investment. In physics, the common difference can be used to calculate the velocity of an object. In engineering, the common difference can be used to calculate the stress on a material.
Conclusion
In conclusion, arithmetic sequences and their common differences are fundamental concepts in mathematics that have numerous real-world applications. Understanding the common difference is essential for solving various problems in mathematics and has numerous practical applications in finance, physics, and engineering.
References
- [1] "Arithmetic Sequences and Series" by Math Open Reference
- [2] "Arithmetic Sequences" by Khan Academy
- [3] "Common Differences" by Wolfram MathWorld
Additional Resources
- [1] "Arithmetic Sequences and Series" by Mathway
- [2] "Arithmetic Sequences" by Purplemath
- [3] "Common Differences" by IXL
Arithmetic Sequences: Q&A ==========================
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. This constant difference is called the common difference.
Q: How do I find the common difference in an arithmetic sequence?
A: To find the common difference in an arithmetic sequence, you can use the formula:
d = a_n - a_(n-1)
where d is the common difference, a_n is the nth term, and a_(n-1) is the (n-1)th term.
Q: What is the formula for the nth term of an arithmetic sequence?
A: The formula for the nth term of an arithmetic sequence is:
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: How do I find the sum of an arithmetic sequence?
A: To find the sum of an arithmetic sequence, you can use the formula:
S_n = n/2 (a_1 + a_n)
where S_n is the sum of the first n terms, n is the term number, a_1 is the first term, and a_n is the nth term.
Q: What is the formula for the sum of an infinite arithmetic sequence?
A: The formula for the sum of an infinite arithmetic sequence is:
S = a_1 / (1 - r)
where S is the sum of the infinite sequence, a_1 is the first term, and r is the common ratio.
Q: How do I find the common ratio in an arithmetic sequence?
A: To find the common ratio in an arithmetic sequence, you can use the formula:
r = d / a_1
where r is the common ratio, d is the common difference, and a_1 is the first term.
Q: What is the relationship between the common difference and the common ratio?
A: The common difference and the common ratio are related by the formula:
d = r(a_1)
where d is the common difference, r is the common ratio, and a_1 is the first term.
Q: How do I determine if a sequence is arithmetic or not?
A: To determine if a sequence is arithmetic or not, you can check if the difference between any two consecutive terms is constant. If it is, then the sequence is arithmetic.
Q: What are some real-world applications of arithmetic sequences?
A: Arithmetic sequences have numerous real-world applications, including finance, physics, and engineering. For example, in finance, the common difference can be used to calculate the interest rate on a loan or investment. In physics, the common difference can be used to calculate the velocity of an object. In engineering, the common difference can be used to calculate the stress on a material.
Q: How do I use arithmetic sequences in real-world problems?
A: To use arithmetic sequences in real-world problems, you can apply the formulas and concepts learned in this article to solve problems in finance, physics, and engineering. For example, you can use the formula for the nth term to calculate the interest rate on a loan or investment, or use the formula for the sum of an infinite arithmetic sequence to calculate the total cost of a project.
Conclusion
In conclusion, arithmetic sequences and their common differences are fundamental concepts in mathematics that have numerous real-world applications. Understanding the common difference and the formulas for the nth term, sum, and common ratio is essential for solving various problems in mathematics and has numerous practical applications in finance, physics, and engineering.
References
- [1] "Arithmetic Sequences and Series" by Math Open Reference
- [2] "Arithmetic Sequences" by Khan Academy
- [3] "Common Differences" by Wolfram MathWorld
Additional Resources
- [1] "Arithmetic Sequences and Series" by Mathway
- [2] "Arithmetic Sequences" by Purplemath
- [3] "Common Differences" by IXL