Finding The 4th Term Of A Sequence Given Partial Sums

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In the fascinating realm of sequences and series, understanding the relationship between a sequence's terms and its partial sums is crucial. Partial sums, essentially the sums of the first n terms of a sequence, offer a unique lens through which we can explore the underlying structure of the sequence itself. This article delves into a specific problem: if we know the formula for the nth partial sum of a sequence a_n, how do we determine a particular term, such as the 4th term, of that sequence? We will explore the fundamental principles, apply them to a concrete example, and discuss the broader implications of this type of problem in mathematics.

Understanding Partial Sums

Before diving into the specifics, let's solidify our understanding of partial sums. The nth partial sum, often denoted as S_n, represents the sum of the first n terms of a sequence. Mathematically, if our sequence is a_1, a_2, a_3,..., then the partial sums are defined as follows:

  • S_1 = a_1
  • S_2 = a_1 + a_2
  • S_3 = a_1 + a_2 + a_3
  • ...
  • S_n = a_1 + a_2 + a_3 + ... + a_n

The key insight here is that each partial sum accumulates the terms of the sequence up to a certain point. This cumulative nature provides a powerful tool for relating the partial sums to the individual terms. Understanding this relationship is key to solving problems where we're given a formula for the partial sum and asked to find a specific term.

The Problem: Finding the 4th Term

Now, let's consider the specific problem at hand. We are given that the nth partial sum of a sequence a_n is given by the formula:

S_n = Σ(2k + 4) from k=1 to n

Our objective is to find the 4th term of the sequence, which we denote as a_4. To achieve this, we'll leverage the relationship between partial sums and individual terms. The most important concept to grasp is that we can extract a specific term by subtracting consecutive partial sums. Specifically:

  • a_1 = S_1
  • a_2 = S_2 - S_1
  • a_3 = S_3 - S_2
  • a_4 = S_4 - S_3

And in general:

  • a_n = S_n - S_(n-1) for n > 1

This formula arises directly from the definition of partial sums. When we subtract S_(n-1) from S_n, we are effectively removing the sum of the first n-1 terms, leaving only the nth term, a_n. This is the fundamental principle we'll use to solve the problem.

Solution: Calculating the 4th Term

To find a_4, we need to calculate S_4 and S_3. Let's start with S_4:

S_4 = Σ(2k + 4) from k=1 to 4 = (2(1) + 4) + (2(2) + 4) + (2(3) + 4) + (2(4) + 4) = 6 + 8 + 10 + 12 = 36*

Now, let's calculate S_3:

S_3 = Σ(2k + 4) from k=1 to 3 = (2(1) + 4) + (2(2) + 4) + (2(3) + 4) = 6 + 8 + 10 = 24*

Finally, we can find a_4 by subtracting S_3 from S_4:

a_4 = S_4 - S_3 = 36 - 24 = 12

Therefore, the 4th term of the sequence is 12. This is the final answer based on our calculations.

Analyzing the Solution and the Sequence

Our calculation reveals that the 4th term of the sequence is 12. But let's take a step back and analyze the sequence itself. The formula for the nth partial sum, S_n = Σ(2k + 4), suggests that the sequence might be arithmetic. To confirm this, let's find the first few terms of the sequence:

  • a_1 = S_1 = 2(1) + 4 = 6
  • a_2 = S_2 - S_1 = (2(1) + 4) + (2(2) + 4) - 6 = 14 - 6 = 8
  • a_3 = S_3 - S_2 = 24 - 14 = 10
  • a_4 = S_4 - S_3 = 36 - 24 = 12

We observe that the sequence is 6, 8, 10, 12,..., which is indeed an arithmetic sequence with a common difference of 2. This observation validates our solution and provides a deeper understanding of the sequence's behavior.

Generalizing the Approach

The method we used to find the 4th term can be generalized to find any term of the sequence, given the formula for the nth partial sum. The key is to remember the relationship:

  • a_n = S_n - S_(n-1)

This formula allows us to express any term of the sequence in terms of the partial sums. This technique is particularly useful when the formula for the partial sum is simpler to work with than a direct formula for the nth term itself. Mastering this technique will greatly enhance your ability to solve sequence-related problems.

Importance in Mathematical Context

Understanding the relationship between sequences and their partial sums is fundamental in various areas of mathematics, including calculus, real analysis, and discrete mathematics. In calculus, for instance, the concept of partial sums is closely related to the definition of infinite series and their convergence. This concept is the foundation for more advanced mathematical concepts.

In real analysis, partial sums play a crucial role in understanding the behavior of sequences and series of real numbers. They are used to define concepts such as convergence, divergence, and the limit of a sequence. In discrete mathematics, sequences and their partial sums are essential tools for modeling and analyzing various discrete phenomena, such as recurrence relations and combinatorial problems. Understanding sequences and partial sums helps bridge multiple mathematical areas.

Conclusion

In this article, we explored the problem of finding a specific term of a sequence given the formula for its nth partial sum. We learned the fundamental relationship between partial sums and individual terms, and we applied this knowledge to find the 4th term of a particular sequence. Furthermore, we analyzed the sequence and recognized it as an arithmetic sequence, which validated our solution. The techniques and concepts discussed here are essential for anyone studying sequences and series, and they have broad applications in various areas of mathematics. The core idea of partial sums is a powerful tool for analyzing sequences.