Finding Term Position In A Sequence Sum Formula Sn = N³ + N² + N + 1
Introduction
In the fascinating world of mathematical sequences, we often encounter problems that challenge our understanding of patterns and relationships between terms. One such intriguing problem involves finding the position of a specific term within a sequence, given a formula for the sum of its first n terms. In this article, we delve into a problem where we are given the sum of the first n terms of a sequence, denoted as Sₙ, and a specific term tₘ. Our mission is to determine the value of m, which represents the position of the term tₘ in the sequence. This exploration will not only solidify our understanding of sequences and series but also enhance our problem-solving skills in mathematics.
Problem Statement
Consider a sequence t₁, t₂, t₃,... where Sₙ represents the sum of the first n terms of the sequence. We are given that Sₙ = n³ + n² + n + 1 and tₘ = 291. The objective is to find the value of m.
Understanding the Problem
At the heart of this problem lies the relationship between the sum of a sequence and its individual terms. The sum of the first n terms, Sₙ, provides a cumulative value, while each term tₙ contributes a specific amount to that sum. To find tₘ, we need to unravel this relationship and isolate the term we are interested in. This involves understanding how the sum changes as we add each new term to the sequence. The key insight here is that the nth term, tₙ, can be found by subtracting the sum of the first (n-1) terms, Sₙ₋₁, from the sum of the first n terms, Sₙ. This fundamental concept will guide our approach to solving the problem.
The Significance of Summation Formulas
Summation formulas, like the one given in this problem (Sₙ = n³ + n² + n + 1), are powerful tools in mathematics. They allow us to express the sum of a series in a concise and elegant manner. These formulas are not just abstract expressions; they encode the underlying pattern of the sequence. By analyzing the structure of the formula, we can gain insights into the behavior of the sequence, such as how quickly the terms grow or whether the sequence converges or diverges. In our specific case, the cubic nature of the formula suggests that the sequence will exhibit a rapid growth in its terms. This information can be valuable in guiding our problem-solving process, helping us to anticipate the magnitude of the terms and the position of a specific term within the sequence.
Solution
Step 1: Finding the General Term
To find the value of m, we first need to determine the general term tₙ of the sequence. As discussed earlier, the nth term can be found by subtracting the sum of the first (n-1) terms from the sum of the first n terms.
tₙ = Sₙ - Sₙ₋₁
Given Sₙ = n³ + n² + n + 1, we need to find Sₙ₋₁. To do this, we simply replace n with (n-1) in the formula for Sₙ:
Sₙ₋₁ = (n-1)³ + (n-1)² + (n-1) + 1
Now, we expand and simplify Sₙ₋₁:
Sₙ₋₁ = (n³ - 3n² + 3n - 1) + (n² - 2n + 1) + (n - 1) + 1 Sₙ₋₁ = n³ - 2n² + 2n
Now we can find tₙ:
tₙ = Sₙ - Sₙ₋₁ = (n³ + n² + n + 1) - (n³ - 2n² + 2n) tₙ = 3n² - n + 1
This is the general formula for the nth term of the sequence. This formula provides a direct way to calculate any term in the sequence, given its position n. The quadratic nature of this formula indicates that the terms of the sequence will grow quadratically, which is a key characteristic of the sequence that we can use to understand its behavior. This step is crucial because it transforms the problem from one involving sums to one involving individual terms, making it easier to find the position of a specific term.
Step 2: Finding m
We are given that tₘ = 291. Now we can use the general term formula we found in Step 1 to solve for m:
tₘ = 3m² - m + 1 = 291
Subtract 291 from both sides to set the equation to zero:
3m² - m - 290 = 0
This is a quadratic equation in the form of am² + bm + c = 0, where a = 3, b = -1, and c = -290. We can solve this equation using the quadratic formula:
m = (-b ± √(b² - 4ac)) / (2a) m = (1 ± √((-1)² - 4 * 3 * -290)) / (2 * 3) m = (1 ± √(1 + 3480)) / 6 m = (1 ± √3481) / 6 m = (1 ± 59) / 6
We have two possible solutions for m:
m₁ = (1 + 59) / 6 = 60 / 6 = 10 m₂ = (1 - 59) / 6 = -58 / 6 ≈ -9.67
Since m represents the position of a term in the sequence, it must be a positive integer. Therefore, we discard the negative solution.
Thus, m = 10.
This step involves the application of algebraic techniques to solve for the unknown variable m. The use of the quadratic formula is a standard method for solving quadratic equations, and it is essential to understand this method to tackle problems of this nature. The process of finding the roots of the quadratic equation is a critical step in determining the possible values of m. However, it is important to consider the context of the problem and discard any solutions that do not make sense in the given scenario. In this case, we discarded the negative solution because the position of a term in a sequence cannot be negative. This highlights the importance of critical thinking and contextual analysis in problem-solving.
Conclusion
In this article, we successfully found the value of m for the sequence given its sum formula and a specific term. We accomplished this by first determining the general term of the sequence and then solving a quadratic equation. The value of m is 10. This problem showcases the interplay between summation formulas, general terms, and algebraic techniques in solving sequence-related problems. Understanding these concepts is crucial for mastering mathematical sequences and series.
Key Takeaways
- The general term of a sequence can be found by subtracting the sum of the first (n-1) terms from the sum of the first n terms.
- Summation formulas provide a concise way to represent the sum of a series.
- Quadratic equations often arise when solving for the position of a term in a sequence.
- It's essential to consider the context of the problem and discard any solutions that don't make sense.
This problem-solving journey underscores the importance of a systematic approach to mathematical problems. By breaking down the problem into smaller, manageable steps, we were able to navigate the complexities and arrive at the correct solution. The process of finding the general term, setting up the equation, and solving for the unknown variable highlights the power of mathematical tools and techniques in unraveling the patterns and relationships within sequences and series. Moreover, the emphasis on critical thinking and contextual analysis reinforces the importance of understanding the underlying concepts and applying them appropriately.
Further Exploration
To further enhance your understanding of sequences and series, consider exploring the following topics:
- Arithmetic and geometric sequences
- Convergence and divergence of series
- Applications of sequences and series in calculus and other fields
By delving deeper into these areas, you will gain a more comprehensive understanding of the rich and fascinating world of mathematical sequences and series. This knowledge will not only enhance your problem-solving skills but also open doors to further exploration and discovery in the realm of mathematics.
