Identifying the Common Difference

Oct 13, 2025 by talynt.org 100 views

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In this problem, we are given the first three terms of an arithmetic sequence: $$11 \frac{1}{2}, 9 \frac{5}{16}, 7 \frac{1}{8}, \ldots$$

Identifying the Common Difference

To find the common difference, we need to subtract each term from the previous term. Let's calculate the difference between the first two terms:

\begin{aligned} d &= (9 \frac{5}{16}) - (11 \frac{1}{2}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{2}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{8}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{1}{4}) \\ &= (9 + \frac{5}{16}) - (11 + \frac{4}{16}) \\ &= (9 + \frac{5}{<br/> **Understanding the Arithmetic Sequence: A Q&A Guide** ===================================================== **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: How do I find the common difference in an arithmetic sequence?** ---------------------------------------------------------------- A: To find the common difference, you need to subtract each term from the previous term. For example, if the first two terms are 9 and 11, the common difference would be 11 - 9 = 2. **Q: What is the formula for finding the nth term of an arithmetic sequence?** ------------------------------------------------------------------------ A: The formula for finding the nth term of an arithmetic sequence is: an = a1 + (n - 1)d where an is the nth term, a1 is the first term, n is the term number, and d is the common difference. **Q: How do I use the formula to find the fifth term of the given arithmetic sequence?** -------------------------------------------------------------------------------- A: To find the fifth term, we need to plug in the values into the formula. The first term (a1) is 11 1/2, the common difference (d) is -1 7/16, and the term number (n) is 5. an = a1 + (n - 1)d = 11 1/2 + (5 - 1)(-1 7/16) = 11 1/2 + 4(-1 7/16) = 11 1/2 + (-7 1/4) = 11 1/2 - 7 1/4 = 4 15/16 **Q: What is the fifth term of the given arithmetic sequence?** --------------------------------------------------------- A: The fifth term of the given arithmetic sequence is 4 15/16. **Q: Why is it important to understand arithmetic sequences?** --------------------------------------------------------- A: Understanding arithmetic sequences is important in many areas of mathematics, including algebra, geometry, and calculus. It is also used in real-world applications such as finance, engineering, and computer science. **Q: How can I apply the concept of arithmetic sequences to my everyday life?** ------------------------------------------------------------------------- A: You can apply the concept of arithmetic sequences to your everyday life by recognizing patterns and relationships between numbers. For example, if you are saving money for a goal, you can use an arithmetic sequence to calculate how much you need to save each month to reach your goal. **Q: What are some common mistakes to avoid when working with arithmetic sequences?** -------------------------------------------------------------------------------- A: Some common mistakes to avoid when working with arithmetic sequences include: * Not checking if the sequence is arithmetic before trying to find the common difference * Not using the correct formula to find the nth term * Not plugging in the correct values into the formula * Not simplifying the expression before finding the final answer **Q: How can I practice and improve my skills in working with arithmetic sequences?** -------------------------------------------------------------------------------- A: You can practice and improve your skills in working with arithmetic sequences by: * Working through examples and exercises in your textbook or online resources * Creating your own arithmetic sequences and finding the common difference and nth term * Using real-world applications to practice working with arithmetic sequences * Asking a teacher or tutor for help if you are struggling with a concept.