Finding Common Difference And First Term Of Arithmetic Sequence With Solutions

Jul 15, 2025 by ADMIN 79 views

Finding the Common Difference and First Term of an Arithmetic Sequence

This comprehensive guide will walk you through the process of finding the common difference and the first term of an arithmetic sequence, given two terms in the sequence. We will explore the underlying concepts, provide a step-by-step solution, and delve into the importance of understanding arithmetic sequences. Understanding arithmetic sequences is crucial in various mathematical and real-world applications, and this guide aims to equip you with the necessary knowledge and skills to tackle these problems effectively. This guide provides a detailed explanation and step-by-step solution for determining the common difference and the first term of an arithmetic sequence, crucial concepts in mathematics with wide-ranging applications.

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, often denoted by 'd'. The first term of the sequence is usually denoted by 'a₁' or 'a'. Each subsequent term in the sequence can be obtained by adding the common difference to the preceding term. Arithmetic sequences are fundamental in mathematics and have applications in various fields, including finance, physics, and computer science. Their predictable nature makes them easy to analyze and model. In essence, an arithmetic sequence is a list of numbers where the difference between any two adjacent numbers is constant. This constant difference, known as the common difference, is the key to understanding and working with these sequences. For instance, consider the sequence 2, 5, 8, 11, 14... Here, the common difference is 3 because each term is obtained by adding 3 to the previous term. Identifying arithmetic sequences and understanding their properties is a basic yet essential skill in mathematics. Arithmetic sequences appear in various real-world scenarios, such as calculating simple interest, modeling linear growth, and even in patterns found in nature. The ability to recognize and analyze arithmetic sequences provides a solid foundation for more advanced mathematical concepts. The general form of an arithmetic sequence can be represented as: a, a + d, a + 2d, a + 3d, ..., where 'a' is the first term and 'd' is the common difference. This formula allows us to find any term in the sequence if we know the first term and the common difference. Understanding the general form is crucial for solving problems involving arithmetic sequences, as it provides a framework for relating different terms in the sequence. The formula for the nth term of an arithmetic sequence is aₙ = a + (n - 1)d, where aₙ is the nth term, a is the first term, n is the term number, and d is the common difference. This formula is a powerful tool for finding specific terms in the sequence without having to list all the terms before it. For example, if we want to find the 100th term of a sequence, we can directly substitute the values into the formula instead of calculating the first 99 terms. Mastering this formula is essential for efficiently solving problems related to arithmetic sequences. Arithmetic sequences are not just abstract mathematical concepts; they have practical applications in various fields. For example, calculating simple interest on a loan or investment involves an arithmetic sequence. Similarly, the depreciation of an asset over time can be modeled using an arithmetic sequence. In computer science, arithmetic sequences are used in algorithms and data structures. Understanding these real-world applications can help appreciate the importance of arithmetic sequences in various disciplines. In summary, arithmetic sequences are sequences with a constant difference between consecutive terms. They are fundamental in mathematics and have numerous real-world applications. Understanding the general form and the formula for the nth term is crucial for working with arithmetic sequences effectively. This foundation will enable you to solve more complex problems involving sequences and series.

Problem Statement

We are given two terms of an arithmetic sequence: the 24th term ( $a_{24}$ ) is 85, and the 28th term ( $a_{28}$ ) is 100. Our task is to find the common difference (d) and the first term (a₁) of this sequence. The challenge lies in utilizing the information from these two terms to deduce the underlying structure of the sequence, specifically the constant difference between terms and the starting point of the sequence. This type of problem is a classic application of arithmetic sequence principles and requires a solid understanding of the relationship between terms in the sequence. By solving this problem, we not only find the specific values of the common difference and the first term but also reinforce our understanding of how arithmetic sequences work in general. This skill is valuable for tackling more complex problems involving sequences and series, as well as for real-world applications where arithmetic patterns occur. The ability to extract information from limited data points and extrapolate the entire sequence is a powerful tool in mathematical problem-solving. This problem serves as an excellent example of how we can use known terms to uncover the underlying rules that govern a sequence. By carefully applying the formulas and concepts related to arithmetic sequences, we can systematically determine the missing pieces of information and gain a complete understanding of the sequence. The problem highlights the importance of precision and attention to detail in mathematical calculations, as a small error in any step can lead to incorrect results. It also emphasizes the need for a clear and logical approach to problem-solving, where each step is justified and contributes to the overall solution. By working through this problem, we develop not only our mathematical skills but also our problem-solving abilities, which are applicable in various contexts beyond mathematics. In essence, this problem is more than just finding two numbers; it's about developing a deeper understanding of arithmetic sequences and honing our mathematical reasoning skills. The given information allows us to set up equations that relate the terms to the common difference and the first term. Solving these equations will lead us to the values we seek. This process involves algebraic manipulation and a systematic approach to problem-solving. Understanding the relationships between terms in an arithmetic sequence is crucial for solving this type of problem. The formula for the nth term of an arithmetic sequence, aₙ = a₁ + (n - 1)d, plays a central role in this solution. By applying this formula to the given terms, we can create a system of equations that can be solved for the unknowns. This problem serves as a practical application of the theoretical concepts of arithmetic sequences and demonstrates the power of mathematical tools in solving real-world problems. The solution process involves careful attention to detail and a systematic approach to algebraic manipulation. By working through this problem, we not only find the answers but also reinforce our understanding of the underlying concepts and techniques. This knowledge will be valuable for tackling more complex problems involving sequences and series.

Solution: Finding the Common Difference

To find the common difference (d), we can use the formula for the nth term of an arithmetic sequence: $a_n = a_1 + (n - 1)d$ . We have two equations based on the given information:

$a_{24} = a_1 + (24 - 1)d = 85$
$a_{28} = a_1 + (28 - 1)d = 100$

Subtracting the first equation from the second, we get:

$(a_1 + 27d) - (a_1 + 23d) = 100 - 85$

$4d = 15$

$d = rac{15}{4} = 3.75$

Therefore, the common difference is 3.75. This approach effectively eliminates the first term (a₁) from the equations, allowing us to isolate and solve for the common difference (d). The subtraction method is a powerful technique for solving systems of linear equations, and it is particularly useful in this context because it simplifies the problem and allows us to focus on the key variable we need to find. The clarity and efficiency of this method make it a valuable tool for solving similar problems involving arithmetic sequences. By understanding this technique, we can approach more complex problems with confidence and solve them systematically. The common difference, 3.75, represents the constant amount by which each term in the sequence increases. This value is crucial for understanding the behavior of the sequence and for finding other terms in the sequence. Knowing the common difference allows us to predict future terms and to analyze the overall trend of the sequence. In the context of real-world applications, the common difference might represent a constant rate of growth, a constant rate of depreciation, or any other linear pattern. The ability to calculate and interpret the common difference is a valuable skill in various fields. The process of subtracting the equations is a key step in solving the problem. This step eliminates the variable a₁, which simplifies the equation and allows us to solve for d directly. The resulting equation, 4d = 15, is a simple linear equation that can be easily solved by dividing both sides by 4. The solution, d = 3.75, represents the constant difference between consecutive terms in the arithmetic sequence. This value is essential for understanding the pattern and behavior of the sequence. Knowing the common difference, we can now proceed to find the first term of the sequence. The common difference is a fundamental characteristic of an arithmetic sequence, and its value provides valuable information about the sequence's properties. It represents the constant increment or decrement between consecutive terms, and it is essential for determining the overall trend and behavior of the sequence. In this case, the common difference of 3.75 indicates that the sequence is increasing, and each term is 3.75 greater than the previous term. This understanding is crucial for predicting future terms in the sequence and for applying the sequence to real-world scenarios. The common difference is not just a numerical value; it is a key parameter that defines the essence of the arithmetic sequence. The process of finding the common difference involves using the given information to set up equations and then solving those equations using algebraic techniques. This process is a fundamental skill in mathematics, and it is applicable to a wide range of problems. The ability to translate a problem into mathematical equations and then solve those equations is a crucial step in mathematical problem-solving. By mastering this skill, we can approach complex problems with confidence and solve them systematically. The solution to this part of the problem, d = 3.75, is a significant step towards understanding the arithmetic sequence. With this value, we can now proceed to find the first term of the sequence and gain a complete understanding of its properties. The common difference is a crucial piece of information that allows us to reconstruct the entire sequence and to predict its behavior.

Solution: Finding the First Term

Now that we have the common difference, we can find the first term ( $a_1$ ) by substituting the value of d into either of the original equations. Let's use the first equation:

$85 = a_1 + 23d$

$85 = a_1 + 23(3.75)$

$85 = a_1 + 86.25$

$a_1 = 85 - 86.25$

$a_1 = -1.25$

Thus, the first term of the arithmetic sequence is -1.25. This step demonstrates the power of using previously calculated results to solve for other unknowns in a problem. By substituting the common difference into one of the original equations, we can isolate the first term and solve for its value. This approach highlights the interconnectedness of mathematical concepts and the importance of building upon previous findings. The accuracy of this step depends on the accuracy of the previously calculated common difference, emphasizing the need for careful attention to detail in each step of the problem-solving process. The first term, -1.25, is the starting point of the arithmetic sequence. It is the value of the sequence when n = 1. Knowing the first term and the common difference allows us to completely define the arithmetic sequence and to find any term in the sequence. The first term is a fundamental characteristic of the sequence, and its value provides valuable information about the sequence's properties. In the context of real-world applications, the first term might represent an initial value, a starting point, or a base level. The ability to calculate and interpret the first term is a valuable skill in various fields. The process of substituting the common difference and solving for the first term involves algebraic manipulation and a systematic approach. This process is a fundamental skill in mathematics, and it is applicable to a wide range of problems. The ability to substitute values into equations and solve for unknowns is a crucial step in mathematical problem-solving. By mastering this skill, we can approach complex problems with confidence and solve them systematically. The solution to this part of the problem, a₁ = -1.25, completes our understanding of the arithmetic sequence. We now know both the first term and the common difference, which allows us to describe the sequence completely. The first term is a crucial piece of information that, along with the common difference, defines the entire sequence. With these two values, we can find any term in the sequence and analyze its behavior. The process of finding the first term reinforces our understanding of the arithmetic sequence formula and its applications. The ability to manipulate the formula and solve for different variables is a key skill in mathematics. The negative value of the first term indicates that the sequence starts below zero and increases with each subsequent term. This understanding provides further insight into the behavior of the sequence and its position on the number line. The first term is a fundamental characteristic of the arithmetic sequence, and its value provides valuable information about the sequence's overall properties. By knowing the first term and the common difference, we can completely describe the sequence and predict its future terms. This knowledge is essential for applying arithmetic sequences to real-world scenarios and for solving more complex problems involving sequences and series. The first term is not just a numerical value; it is a key parameter that defines the starting point of the sequence and its overall behavior. The successful determination of the first term marks the completion of the problem and demonstrates a thorough understanding of arithmetic sequences and their properties. The ability to find both the common difference and the first term showcases a mastery of the concepts and techniques involved in solving problems related to arithmetic sequences.

Conclusion

We have successfully found the common difference (d = 3.75) and the first term ( $a_1 = -1.25$ ) of the arithmetic sequence. This demonstrates the application of the arithmetic sequence formula and algebraic manipulation to solve problems involving sequences. In conclusion, this problem-solving exercise has reinforced the importance of understanding arithmetic sequences and their properties. By systematically applying the formula for the nth term and using algebraic techniques, we were able to determine the common difference and the first term of the sequence. This skill is valuable for tackling more complex problems involving sequences and series, as well as for real-world applications where arithmetic patterns occur. The ability to extract information from limited data points and extrapolate the entire sequence is a powerful tool in mathematical problem-solving. This problem serves as an excellent example of how we can use known terms to uncover the underlying rules that govern a sequence. By carefully applying the formulas and concepts related to arithmetic sequences, we can systematically determine the missing pieces of information and gain a complete understanding of the sequence. The problem highlights the importance of precision and attention to detail in mathematical calculations, as a small error in any step can lead to incorrect results. It also emphasizes the need for a clear and logical approach to problem-solving, where each step is justified and contributes to the overall solution. By working through this problem, we developed not only our mathematical skills but also our problem-solving abilities, which are applicable in various contexts beyond mathematics. The solution process involved a combination of theoretical knowledge and practical application. We started by understanding the fundamental concepts of arithmetic sequences, including the definition of the common difference and the formula for the nth term. We then applied these concepts to the given problem, setting up equations based on the provided information. Finally, we used algebraic techniques to solve the equations and find the unknown values. This process demonstrates the importance of having a solid foundation in mathematical theory and the ability to apply that theory to real-world problems. The successful completion of this problem showcases a mastery of the concepts and techniques related to arithmetic sequences. The ability to find both the common difference and the first term is a crucial skill for solving a wide range of problems involving sequences and series. This skill is not only valuable in mathematics but also in various other fields, such as finance, physics, and computer science. Arithmetic sequences are a fundamental concept in mathematics, and a thorough understanding of them is essential for further studies in mathematics and related fields. The concepts and techniques learned in this problem can be applied to more complex problems involving arithmetic sequences and series. For example, we can use these techniques to find the sum of a finite arithmetic series or to determine the number of terms in a sequence that meet certain criteria. The ability to solve problems involving arithmetic sequences is a valuable asset in any mathematical toolkit. This exercise demonstrates the importance of a systematic approach to problem-solving. By breaking down the problem into smaller, manageable steps, we were able to solve it effectively. This approach can be applied to a wide range of problems in mathematics and other fields. The key steps in our solution were: understanding the problem, identifying the relevant concepts and formulas, setting up equations, solving the equations, and interpreting the results. By following these steps, we can approach complex problems with confidence and solve them systematically. The success we achieved in solving this problem reinforces the importance of perseverance and attention to detail. Mathematical problem-solving often requires patience and persistence, as well as careful attention to detail to avoid errors. By working through this problem, we have developed not only our mathematical skills but also our problem-solving abilities, which are applicable in various contexts. The solutions we found, d = 3.75 and a₁ = -1.25, provide a complete description of the arithmetic sequence. We now know the starting point of the sequence and the constant difference between terms. This knowledge allows us to predict any term in the sequence and to analyze its behavior. The ability to describe and analyze arithmetic sequences is a valuable skill in mathematics and its applications. The arithmetic sequence formula and algebraic manipulation are powerful tools for solving problems involving sequences and series. By mastering these tools, we can approach a wide range of problems with confidence and solve them effectively. The knowledge and skills we have gained in this exercise will be valuable for further studies in mathematics and related fields. This problem-solving experience has been a valuable learning opportunity, and it has reinforced our understanding of arithmetic sequences and their properties. By continuing to practice and apply these concepts, we can further develop our mathematical skills and problem-solving abilities. The successful completion of this problem is a testament to our ability to learn and apply mathematical concepts.