General Term And Sum Of Series 1*n + 2*(n-1) + 3*(n-2) +
In the realm of mathematical sequences and series, a fascinating challenge arises when we encounter patterns that require us to delve deeper than simple arithmetic or geometric progressions. One such intriguing series is: 1·n + 2·(n-1) + 3·(n-2) + ... This series presents a unique structure where each term is a product of two factors, one increasing linearly and the other decreasing linearly. To fully understand and work with this series, we need to find the general term and then derive a formula for the sum of its first n terms. This exploration will not only enhance our understanding of series but also sharpen our skills in algebraic manipulation and summation techniques.
Identifying the General Term (Tₖ)
To embark on our journey, let's first focus on identifying the general term, often denoted as Tₖ, of the series. Understanding the general term is crucial as it encapsulates the pattern of the series and allows us to express any term in the sequence. Analyzing the given series, 1·n + 2·(n-1) + 3·(n-2) + ..., we can observe a distinct pattern. The first factor in each term increments by 1, while the second factor decrements by 1. This interplay between increasing and decreasing factors is the hallmark of this series.
Let's dissect this pattern more formally. If we consider the k-th term of the series, we can see that the first factor is simply k. The second factor, on the other hand, can be expressed as (n - k + 1). This is because when k = 1, the second factor is n; when k = 2, it's n-1; and so on. Therefore, the k-th term, Tₖ, can be written as:
Tₖ = k * (n - k + 1)
This elegantly captures the essence of the series. Expanding this expression, we get:
Tₖ = k * (n - k + 1) = nk - k² + k
This form of the general term will be particularly useful when we move on to calculating the sum of the series. The ability to express the general term in a compact algebraic form is a powerful tool in dealing with series. It allows us to use summation formulas and techniques to find the sum of any number of terms.
Now that we have successfully identified the general term, our next step is to use it to find the sum of the first n terms of the series. This will involve using summation notation and applying some algebraic manipulation to arrive at a closed-form expression for the sum. This expression will then allow us to calculate the sum of the series for any given value of n.
Deriving the Sum of the First n Terms (Sₙ)
With the general term Tₖ = nk - k² + k firmly established, we can now proceed to the second part of our challenge: finding the sum of the first n terms of the series. This sum is typically denoted as Sₙ and represents the sum of the first n terms of the series. To find Sₙ, we will utilize summation notation and leverage some well-known summation formulas. The sum of the first n terms, Sₙ, is a crucial characteristic of a series, providing a concise way to represent the total value of a finite segment of the series.
Using summation notation, we can express Sₙ as follows:
Sₙ = Σ(Tₖ) from k = 1 to n
Substituting our expression for Tₖ, we get:
Sₙ = Σ(nk - k² + k) from k = 1 to n
Now, we can use the properties of summation to split this into three separate summations:
Sₙ = Σ(nk) from k = 1 to n - Σ(k²) from k = 1 to n + Σ(k) from k = 1 to n
Each of these summations has a known formula. Let's recall these formulas:
- Σ(k) from k = 1 to n = n(n + 1) / 2
- Σ(k²) from k = 1 to n = n(n + 1)(2n + 1) / 6
- Σ(nk) from k = 1 to n = n * Σ(k) from k = 1 to n = n * [n(n + 1) / 2] = n²(n + 1) / 2
Substituting these formulas into our expression for Sₙ, we get:
Sₙ = [n²(n + 1) / 2] - [n(n + 1)(2n + 1) / 6] + [n(n + 1) / 2]
Now, we need to simplify this expression. To do this, we'll find a common denominator (6) and combine the terms:
Sₙ = [3n²(n + 1) - n(n + 1)(2n + 1) + 3n(n + 1)] / 6
We can factor out n(n + 1) from each term in the numerator:
Sₙ = n(n + 1) * [3n - (2n + 1) + 3] / 6
Simplifying the expression inside the brackets, we get:
Sₙ = n(n + 1) * [3n - 2n - 1 + 3] / 6 Sₙ = n(n + 1) * (n + 2) / 6
Therefore, the sum of the first n terms of the series is:
Sₙ = n(n + 1)(n + 2) / 6
This is a closed-form expression for Sₙ, meaning that we can directly calculate the sum of the first n terms by simply substituting the value of n into this formula. This closed-form expression is a powerful result, providing a concise and efficient way to calculate the sum of the series. It eliminates the need to add up each term individually, especially when dealing with a large number of terms.
Conclusion: A Journey Through Series and Summation
In this exploration, we successfully navigated the series 1·n + 2·(n-1) + 3·(n-2) + ... to find both its general term and the sum of its first n terms. We began by carefully analyzing the series to identify the pattern in its terms, which led us to the general term Tₖ = k * (n - k + 1) = nk - k² + k. This general term served as the foundation for our next step: finding the sum of the first n terms.
To find the sum, Sₙ, we utilized summation notation and properties to express the sum as a combination of simpler summations. By recalling and applying the standard summation formulas for Σ(k) and Σ(k²), we were able to derive a closed-form expression for Sₙ. After careful algebraic manipulation and simplification, we arrived at the elegant result:
Sₙ = n(n + 1)(n + 2) / 6
This journey highlights the power of mathematical tools and techniques in unraveling the mysteries of series. Understanding the general term and deriving a formula for the sum are fundamental skills in working with series and sequences. These skills are not only valuable in mathematics but also in various fields that involve pattern recognition and quantitative analysis.
By mastering these techniques, we can confidently tackle a wide range of series problems and gain a deeper appreciation for the beauty and structure of mathematical sequences. The process of finding the general term and sum of a series is a testament to the power of observation, algebraic manipulation, and the application of established mathematical principles. This exploration serves as a valuable exercise in mathematical problem-solving and reinforces the importance of a systematic approach to complex problems.
In conclusion, the series 1·n + 2·(n-1) + 3·(n-2) + ... provides a compelling example of how we can use mathematical tools to understand and analyze patterns in sequences. The general term, Tₖ = k * (n - k + 1), and the sum of the first n terms, Sₙ = n(n + 1)(n + 2) / 6, represent key insights into the structure and behavior of this series. This exploration not only enhances our mathematical skills but also demonstrates the elegance and power of mathematical reasoning.
