General Term And Sum Of Series 1*n + 2*(n-1) + 3*(n-2)

Understanding the Series

To find the general term and the sum of the first n terms of the given series, we must first deeply understand the pattern and structure of the series itself. The series presented is: 1⋅n + 2⋅(n-1) + 3⋅(n-2) + .... At first glance, it appears to be a sum of products, where each product involves two factors. The first factor in each term increases sequentially (1, 2, 3, ...), while the second factor decreases sequentially from n. This interplay between increasing and decreasing sequences is key to unlocking the general term. Analyzing the components of each term, we see that the k-th term (where k represents the position of the term in the series) can be expressed as the product of k and (n - k + 1). The first factor, k, is straightforward and represents the natural number sequence. The second factor, (n - k + 1), requires more attention. When k = 1, the second factor is n; when k = 2, it is n - 1; and so on. This confirms the decreasing sequence pattern. Therefore, the k-th term captures the essence of the series' structure, laying the groundwork for further analysis. This foundational understanding is crucial because it allows us to generalize the series and express it in a compact, mathematical form. Without this initial decomposition and analysis, proceeding to find the sum of the series would be significantly more challenging. Understanding the series not only helps in identifying the general term but also paves the way for employing summation techniques to determine the sum of the first n terms.

Deriving the General Term

The general term of a series is the formula that represents any term in the series based on its position. In our case, having analyzed the series 1⋅n + 2⋅(n-1) + 3⋅(n-2) + ..., we can now formally derive this general term. As we observed earlier, the k-th term in the series is formed by the product of the term's position (k) and a factor that decreases sequentially from n. Mathematically, this decreasing factor can be expressed as (n - k + 1). Thus, the k-th term, which we'll denote as T_k, is given by the product of k and (n - k + 1). Expressing this mathematically, we get: T_k = k(n - k + 1). This formula encapsulates the essence of the series' construction. To further refine the general term, we can expand the expression: T_k = nk - k² + k. This expanded form is particularly useful when we aim to compute the sum of the series, as it separates the term into components that are easier to sum using standard summation formulas. The derived general term, T_k = nk - k² + k, is a pivotal result. It allows us to express any term in the series solely based on its position k and the given parameter n. This ability to represent individual terms through a single formula is a cornerstone of series analysis, enabling us to move forward with finding the sum of the first n terms. Without this general term, calculating the sum would be an arduous task, requiring the manual addition of each term. Therefore, the derived general term not only simplifies the representation of the series but also provides the necessary foundation for summing its terms efficiently. Understanding the derivation and significance of the general term is crucial for anyone studying series and sequences.

Calculating the Sum of the First n Terms

After deriving the general term T_k = nk - k² + k, we can now focus on calculating the sum of the first n terms of the series. The sum, which we'll denote as S_n, is the summation of T_k from k = 1 to n. Mathematically, this is expressed as: S_n = Σ(T_k) from k = 1 to n. Substituting the general term into the summation, we get: S_n = Σ(nk - k² + k) from k = 1 to n. To simplify this summation, we can break it down into three separate summations, each corresponding to a term in the general expression: S_n = nΣ(k) - Σ(k²) + Σ(k) from k = 1 to n. Here, we've used the linearity property of summation, which allows us to distribute the summation over addition and subtraction and factor out constants. Now, we can apply standard summation formulas for Σ(k) and Σ(k²). The sum of the first n natural numbers, Σ(k), is given by n(n + 1) / 2. The sum of the squares of the first n natural numbers, Σ(k²), is given by n(n + 1)(2n + 1) / 6. Substituting these formulas into our expression for S_n, we get: S_n = n[n(n + 1) / 2] - [n(n + 1)(2n + 1) / 6] + [n(n + 1) / 2]. This expression can be further simplified by finding a common denominator and combining the terms. S_n = [3n²(n + 1) - n(n + 1)(2n + 1) + 3n(n + 1)] / 6. Factoring out the common term n(n + 1), we have: S_n = n(n + 1)[3n - (2n + 1) + 3] / 6. Simplifying the terms inside the brackets, we get: S_n = n(n + 1)(n + 2) / 6. This final expression represents the sum of the first n terms of the given series. The sum S_n = n(n + 1)(n + 2) / 6 showcases the elegance and power of mathematical summation techniques. It provides a compact formula to calculate the sum of any number of terms in the series, without needing to add each term individually. This result is a testament to the effectiveness of breaking down a complex problem into smaller, manageable parts and applying known formulas and principles. Understanding the steps involved in deriving this sum is not only crucial for solving this specific problem but also for tackling similar series and sequences problems in mathematics.

Example Application

To solidify our understanding of the derived general term and the sum of the first n terms, let's apply them to a specific example. Suppose we want to find the sum of the first 5 terms of the series: 1⋅n + 2⋅(n-1) + 3⋅(n-2) + ... when n = 5. First, let's verify our general term formula, T_k = nk - k² + k. We can list the first few terms of the series when n = 5: 1⋅5, 2⋅4, 3⋅3, 4⋅2, 5⋅1, which simplifies to 5, 8, 9, 8, 5. Now, let's use our general term formula to calculate these terms. For k = 1, T₁ = 5⋅1 - 1² + 1 = 5. For k = 2, T₂ = 5⋅2 - 2² + 2 = 10 - 4 + 2 = 8. For k = 3, T₃ = 5⋅3 - 3² + 3 = 15 - 9 + 3 = 9. For k = 4, T₄ = 5⋅4 - 4² + 4 = 20 - 16 + 4 = 8. For k = 5, T₅ = 5⋅5 - 5² + 5 = 25 - 25 + 5 = 5. The results match the terms we listed, confirming the validity of our general term formula. Next, let's use our sum formula, S_n = n(n + 1)(n + 2) / 6, to calculate the sum of the first 5 terms when n = 5. Substituting n = 5 into the formula, we get: S₅ = 5(5 + 1)(5 + 2) / 6 = 5⋅6⋅7 / 6 = 35. Therefore, the sum of the first 5 terms of the series when n = 5 is 35. We can also manually add the terms we calculated earlier (5 + 8 + 9 + 8 + 5) to verify our result. The manual sum is indeed 35, which further validates our sum formula. This example demonstrates the practical application of the derived formulas. It shows how the general term can be used to find any term in the series, and how the sum formula provides a quick and efficient way to calculate the sum of the first n terms. By working through this example, we gain confidence in our understanding and ability to apply these mathematical concepts. Such examples are crucial for mastering series and sequences, as they bridge the gap between theory and application.

Conclusion

In conclusion, we've successfully found the general term and the sum of the first n terms of the series 1⋅n + 2⋅(n-1) + 3⋅(n-2) + .... We began by carefully analyzing the series to identify its underlying pattern, which led us to derive the general term, T_k = nk - k² + k. This general term is a concise formula that represents any term in the series based on its position k and the parameter n. Following the derivation of the general term, we moved on to calculate the sum of the first n terms. By applying the principles of summation and utilizing standard summation formulas for natural numbers and their squares, we arrived at the sum formula: S_n = n(n + 1)(n + 2) / 6. This formula provides a direct and efficient way to calculate the sum of the series for any given value of n, without the need to manually add each term. To solidify our understanding, we worked through a specific example where we calculated the sum of the first 5 terms when n = 5. We verified our formulas by manually calculating the terms and their sum, confirming the accuracy of our derived formulas. The process of finding the general term and the sum of a series is a fundamental concept in mathematics, particularly in the study of sequences and series. It demonstrates the power of mathematical analysis and the elegance of mathematical formulas in representing complex patterns. The techniques and principles used in this problem are applicable to a wide range of similar problems, making this exercise a valuable learning experience. Mastering these concepts not only enhances mathematical skills but also fosters problem-solving abilities that are applicable in various fields. The ability to identify patterns, derive general formulas, and apply them to solve specific problems is a hallmark of mathematical thinking and a valuable asset in both academic and professional pursuits. Therefore, a thorough understanding of series and sequences, including the methods for finding general terms and sums, is essential for anyone seeking a strong foundation in mathematics.