Problem 145 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the sequence. Then find an expression for the \(n\) th partial sum. $$a_{n}=\frac{1}{n+1}-\frac{1}{n+2}$$

Step-by-Step Solution

Answer

The first five terms of the sequence are: \(\frac{1}{6}\), \(\frac{1}{12}\), \(\frac{1}{20}\), \(\frac{1}{30}\), \(\frac{1}{42}\). The partial sum \(S_{n}\) of the first \(n\) terms is \(S_{n} = 1-\frac{1}{n+2}\)

Step 1: Find the first five terms

Substitute \(n\) equals 1, then 2, 3, 4 and 5 to get the first five terms of the sequence. Given is \(a_{n}=\frac{1}{n+1}-\frac{1}{n+2}\). By substituting the values you get: For \(n=1\), \(a_{1}=\frac{1}{2}-\frac{1}{3} = \frac{1}{6}\). For \(n=2\), \(a_{2}=\frac{1}{3}-\frac{1}{4} = \frac{1}{12}\). For \(n=3\), \(a_{3}=\frac{1}{4}-\frac{1}{5} = \frac{1}{20}\). For \(n=4\), \(a_{\4}=\frac{1}{5}-\frac{1}{6} = \frac{1}{30}\). And finally, for \(n=5\), \(a_{5}=\frac{1}{6}-\frac{1}{7} = \frac{1}{42}\)

Step 2: Find the sum of the sequence

The partial sum \(S_{n}\) of the first \(n\) terms can be found by summing all terms up to the \(n\) th term. We see a pattern where terms are cancelling each other out. Therefore, the sum of the first \(n\) terms (\(n>1\)) simplifies to \(S_{n} = 1-\frac{1}{n+2}\)

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a series of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. For instance, in the sequence 2, 4, 6, 8, each term increases by 2, which is the common difference. To identify if a sequence is arithmetic, we look for this consistent increment or decrement between terms.

However, not all sequences with a simple rule are arithmetic. The sequence given in the exercise, for instance, does not have a constant difference between terms. Instead, each term is defined by the expression \(a_{n}=\frac{1}{n+1}-\frac{1}{n+2}\), which does not result in a fixed difference. Therefore, this sequence is not arithmetic, highlighting the importance of understanding the defining formula of a sequence to properly categorize it.

Sequence Term Calculation

Calculating the terms of a sequence involves finding the value of each term based on its position (n) in the sequence. For the given sequence \(a_{n}=\frac{1}{n+1}-\frac{1}{n+2}\), we substitute values of \(n\) to find the terms. This process reveals the unique value of each term according to its ordinal position.

For practical applications, it's crucial not just to compute individual terms, but to discern the pattern of the sequence – something we analyze by calculating multiple terms, as shown in the step-by-step solution. Understanding this pattern can lead to recognizing more complex properties of the sequence, such as convergence or the behavior of its partial sums.

Summing Series

The sum of a series, often referred to as the series' partial sum, denotes the addition of the terms up to a certain point. It is crucial in many areas of mathematics, including calculus and financial theory. Particularly for a sequence where terms cancel each other out when summed, like the one in our exercise, the calculation of a partial sum becomes straightforward once the cancellation pattern is recognized.

For sequences with simple rules, summing might involve a formula linking the sum to the number of terms summed. The cancellation observed in the sequence from our problem simplifies the process of summing the series, where, after cancellation, the partial sum becomes \(S_{n} = 1-\frac{1}{n+2}\). Recognizing such cancellation patterns drastically reduces computational complexity.

Convergent Series

A convergent series is one in which the partial sums tend towards a specific limit as more and more terms are included. This differentiates them from divergent series where the sums do not approach a single value. Determining convergence is essential in understanding the long-term behavior of series, particularly in contexts such as infinite series and their application in real-world scenarios.

The sequence provided in the exercise does not directly address convergence, but analyzing the pattern of partial sums can lead to conclusions about the nature of the series. As the number of terms increases, the partial sum \(S_{n} = 1-\frac{1}{n+2}\) approaches 1, suggesting that the series is convergent, and its limit as \(n\) approaches infinity would be 1. Understanding convergence is foundational to the study of series and integral to subjects such as analysis and mathematical modeling.