Problem 143 - 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. $$a_{n}=\frac{1}{2 n}-\frac{1}{2 n+2}$$

Step-by-Step Solution

Answer

First five terms of the sequence are \(\frac{1}{3}, \frac{1}{8}, \frac{1}{15}, \frac{1}{24}, \frac{1}{35}\) and the \(n\)th partial sum of the sequence is \(\frac{1}{2}-\frac{1}{2n+2}\).

Step 1: Calculate First Five Terms

We substitute the first five natural numbers for \(n\) in \(a_{n}=\frac{1}{2n}-\frac{1}{2n+2}\) to find the first five terms of the sequence. These are:\(a_1 = \frac{1}{2(1)} - \frac{1}{2(1 + 2)} = \frac{1}{2} - \frac{1}{6} = \frac{1}{3}\),\(a_2 = \frac{1}{2(2)} - \frac{1}{2(2 + 2)} = \frac{1}{4} - \frac{1}{8} = \frac{1}{8}\),\(a_3 = \frac{1}{2(3)} - \frac{1}{2(3 + 2)} = \frac{1}{6} - \frac{1}{10} = \frac{1}{15}\),\(a_4 = \frac{1}{2(4)} - \frac{1}{2(4 + 2)} = \frac{1}{8} - \frac{1}{12} = \frac{1}{24}\),\(a_5 = \frac{1}{2(5)} - \frac{1}{2(5 + 2)} = \frac{1}{10} - \frac{1}{14} = \frac{1}{35}\).

Step 2: Find Sum Expression

To find the sum, look at the first few terms: \(\frac{1}{3}, \frac{1}{8}, \frac{1}{15}, \frac{1}{24}, \frac{1}{35}\). Notice the pattern: the sum of each term is \(\frac{1/2} - \frac{1}{2n+2}\). So, the \(n\)th partial sum \(S_n\) will be the sum of these terms for all values of \(n\). This gives us:\(S_n = \sum_{k=1}^{n} a_k = \sum_{k=1}^{n} (\frac{1}{2k} - \frac{1}{2k+2})\)Simplify this to:\(S_n = \frac{1}{2} + \frac{1}{4} + ... + \frac{1}{2n} - (\frac{1}{4} + \frac{1}{6} + ... + \frac{1}{2n+2})\)Cancelling terms:\(S_n = \frac{1}{2} - \frac{1}{2n+2}\)

Key Concepts

Partial Sum

In mathematics, a partial sum is the sum of the first few terms of an infinite series. Instead of summing all terms indefinitely, we stop at a certain point, giving us what's called the "partial sum." This is useful in understanding the behavior of a series, allowing us to see how the sum grows as more terms are included. For the sequence given in the exercise, the partial sum is represented as the sum of the first \( n \) terms. The formula for the partial sum, denoted as \( S_n \), is calculated as:

• \( S_n = \sum_{k=1}^{n} a_k = \sum_{k=1}^{n} \left( \frac{1}{2k} - \frac{1}{2k+2} \right) \)

This represents the cumulative value when each term of the sequence is combined successively until the nth term. Understanding partial sums can help in approximating the full sum of an infinite series.

Sequence Terms

Sequence terms are individual elements of a sequence, where each element follows a specific formula or pattern. In our case, each term of the sequence is determined by the formula \( a_{n} = \frac{1}{2n} - \frac{1}{2n+2} \). A sequence can be infinite, but sometimes we analyze just the first few terms. Here, these terms are:

• First term, \( a_1 = \frac{1}{3} \)
• Second term, \( a_2 = \frac{1}{8} \)
• Third term, \( a_3 = \frac{1}{15} \)
• Fourth term, \( a_4 = \frac{1}{24} \)
• Fifth term, \( a_5 = \frac{1}{35} \)

Each term follows a reduction based on the simplification of the given expression. Recognizing and calculating these terms is critical to formulating expressions for further mathematical analysis like sums.

Telescoping Series

A telescoping series is a series whose partial sums ultimately only have a few terms left after cancellation. This makes it easier to find the sum since many of the middle terms cancel out. In our exercise, the sequence \( \frac{1}{2n} - \frac{1}{2n+2} \) forms a telescoping series when you add up terms because:

• The positive part \( \frac{1}{2k} \) cancels with negative parts from subsequent terms.

When you add the sequence terms from \( k = 1 \) to \( n \), much of the series will cancel out, greatly simplifying the calculation, resulting in a sum expression \( S_n = \frac{1}{2} - \frac{1}{2n+2} \). Telescoping makes solving complex formulas much more manageable.

Sum Expression

The sum expression is a mathematical expression representing the total of a series. Once the sequence terms and the partial sums have been understood, the sum expression helps evaluate the accumulated value of the sequence up to a certain term.In the current context, after simplifying a telescoping series, we find that the sum expression for partial sums simplifies as:

• \( S_n = \frac{1}{2} - \frac{1}{2n+2} \)

This expression highlights how the overall sum of the series depends only on the initial and one of the subsequent terms due to the telescoping nature of the sequence. It provides a neat and concise representation of what would otherwise be a complicated series of additions.