Arithmetic Series Sum And Term Identification Problems Explained

Introduction

In the realm of mathematics, arithmetic sequences hold a fundamental position, offering a structured and predictable pattern of numbers. Delving into arithmetic sequences unveils a world of fascinating properties and applications, from simple calculations to complex problem-solving scenarios. This article aims to explore two intriguing aspects of arithmetic sequences: calculating the sum of a finite number of terms and identifying the position of a specific term within the sequence. Through clear explanations, step-by-step solutions, and practical examples, we will unravel the mysteries of arithmetic sequences and empower you to master these essential mathematical concepts.

Problem 8 Finding the Sum of the First 10 Terms

In this problem, we embark on a journey to determine the sum of the initial 10 terms of an arithmetic sequence. An arithmetic sequence is a series of numbers where the difference between any two consecutive terms remains constant. This constant difference is known as the common difference. To solve this problem, we are provided with the first term, which is 5, and the last term, which is 40. Our mission is to find the sum of the first 10 terms of this sequence. This can be achieved using the formula for the sum of an arithmetic series, which elegantly connects the first term, the last term, and the number of terms. We will delve into the intricacies of this formula, unraveling its underlying logic and demonstrating its application in this specific scenario. By carefully substituting the given values into the formula, we will arrive at the solution, unveiling the sum of the first 10 terms of this arithmetic sequence. This problem not only provides a numerical answer but also sheds light on the power and elegance of mathematical formulas in solving real-world problems. Understanding the formula for the sum of an arithmetic series is crucial for anyone seeking to master arithmetic sequences and their applications. This knowledge equips us to efficiently calculate the sum of any finite number of terms in an arithmetic sequence, regardless of the complexity of the sequence or the number of terms involved. Furthermore, this problem serves as a stepping stone towards exploring more advanced concepts related to arithmetic sequences and series, such as infinite arithmetic series and their convergence properties. As we delve deeper into the world of mathematics, the ability to confidently manipulate arithmetic sequences and series will prove invaluable in tackling a wide range of mathematical challenges. The exploration of this problem not only enhances our computational skills but also fosters a deeper appreciation for the beauty and order inherent in the world of numbers. By unraveling the mysteries of arithmetic sequences, we unlock a powerful tool for understanding and solving mathematical problems across various domains.

To find the sum of the first 10 terms, we can use the formula for the sum of an arithmetic series:

$$S_n = \frac{n}{2}(a_1 + a_n)$$

Where:

• $S_n$ is the sum of the first n terms
• $n$ is the number of terms
• $a_1$ is the first term
• $a_n$ is the last term

In this case, we have:

• $n = 10$
• $a_1 = 5$
• $a_{10} = 40$

Plugging these values into the formula, we get:

$$S_{10} = \frac{10}{2}(5 + 40) = 5(45) = 225$$

Therefore, the sum of the first 10 terms is 225.

The correct answer is B. 225.

Problem 9 Finding the Term Number for 110

This problem presents us with an arithmetic sequence and challenges us to identify the position of a specific term, 110, within that sequence. The given arithmetic sequence is 12, 19, 26, 33, 40, and so on. To find the term number corresponding to 110, we need to employ the formula for the nth term of an arithmetic sequence. This formula establishes a clear relationship between the nth term, the first term, the common difference, and the term number itself. By carefully analyzing the given sequence, we can determine the first term and the common difference. The first term is simply the initial value of the sequence, while the common difference is the constant amount added to each term to obtain the next term. Once we have identified these key parameters, we can substitute them, along with the target term 110, into the formula for the nth term. This will result in an equation that we can solve for n, the term number. The solution to this equation will reveal the position of 110 within the sequence. This problem highlights the importance of understanding the fundamental properties of arithmetic sequences and the ability to apply the relevant formulas effectively. The formula for the nth term serves as a powerful tool for analyzing and predicting the behavior of arithmetic sequences. By mastering this formula, we can readily determine any term in the sequence, given its position, or conversely, find the position of a specific term within the sequence. This problem also underscores the connection between algebra and arithmetic sequences. The process of solving for n involves algebraic manipulation, demonstrating the interplay between these two branches of mathematics. As we progress in our mathematical journey, the ability to seamlessly integrate concepts from different areas will become increasingly crucial. This problem provides a valuable opportunity to hone our algebraic skills while simultaneously deepening our understanding of arithmetic sequences. The solution to this problem not only yields a numerical answer but also reinforces the importance of careful analysis, logical reasoning, and the effective application of mathematical formulas. By embracing these principles, we can confidently tackle a wide range of mathematical challenges.

To find which term is 110, we first need to determine the common difference (d) of the sequence. This is the difference between any two consecutive terms. In this case:

$$d = 19 - 12 = 7$$

The formula for the nth term ($a_n$) 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
• $d$ is the common difference

We want to find n when $a_n = 110$. We know $a_1 = 12$ and $d = 7$. Plugging these values into the formula, we get:

$$110 = 12 + (n - 1)7$$

Now, we solve for n:

$$110 = 12 + 7n - 7$$

$$110 = 5 + 7n$$

$$105 = 7n$$

$$n = 15$$

Therefore, 110 is the 15th term in the sequence.

The correct answer is B. $15^{\text {th }}$.

Conclusion

Through the exploration of these two problems, we have gained a deeper understanding of arithmetic sequences and their properties. We have successfully calculated the sum of a finite number of terms using the arithmetic series formula and identified the position of a specific term within a sequence using the formula for the nth term. These skills are essential for navigating the world of mathematics and tackling various problem-solving scenarios. By mastering these concepts, we not only enhance our mathematical proficiency but also develop critical thinking and analytical skills that are valuable in all aspects of life. The journey through arithmetic sequences has unveiled the beauty and order inherent in mathematical patterns, fostering a deeper appreciation for the elegance of this discipline.