Problem 146 - 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. $$a_{n}=\frac{1}{n}-\frac{1}{n+2}$$

Step-by-Step Solution

Answer

The first five terms of the sequence are \(1/3, 1/6, 1/12, 1/20, 1/30\). The formula for the nth partial sum of the sequence is \(S_n = 1 + 1/2 + 1/3 - 1/(n+1) - 1/(n+2)\).

Step 1: Compute the first five terms

To find the first five terms, substitute the values of \(n=1, 2, 3, 4, 5\) into the sequence formula \(a_{n}=\frac{1}{n}-\frac{1}{n+2}\). The first five terms are \(a_{1}=\frac{1}{3}\), \(a_{2}=\frac{1}{6}\), \(a_{3}=\frac{1}{12}\), \(a_{4}=\frac{1}{20}\), \(a_{5}=\frac{1}{30}\).

Step 2: Derive the formula for the nth partial sum

The nth partial sum \(S_{n}\) of a sequence is the sum of the first n terms. Here, the given sequence is in the form of a partial fraction, meaning that each term \(a_{n}\) can be written as a difference of two fractions. Therefore, the nth partial sum can be written as:\[S_n = \sum_{k=1}^{n} a_k = (1 - 1/3) + (1/2 - 1/4) + (1/3 - 1/5) + ... + (1/n - 1/(n+2))\]This simplifies to:\[S_n = 1 + 1/2 + 1/3 - 1/(n+1) - 1/(n+2)\]By grouping the positive terms, and the negative terms.

Key Concepts

Sequences in Algebra

In algebra, a sequence is a list of numbers that follow a certain rule. Each number in the sequence is called a term, and the rule that defines the sequence is often expressed in the form of a formula. The term's position within the sequence is denoted by an index, usually represented by the letter 'n'. In the exercise, we are given the sequence defined by the formula

\( a_{n} = \frac{1}{n} - \frac{1}{n+2} \), which generates a list of terms when we input consecutive values of 'n'. Sequences can be finite or infinite, and understanding how to represent and manipulate them is crucial for solving problems in algebra and calculus.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex fractions into simpler, component fractions that are easier to work with, especially in integration and finding series sums. The given sequence represents a difference of two terms, each of which is a fraction. To decompose a fraction like \( \frac{1}{n} - \frac{1}{n+2} \), we can consider each term separately. This method is vital in finding the nth partial sum, as it enables us to cancel terms conveniently when adding the fractions sequentially.

By understanding partial fraction decomposition, we can simplify complex expressions and solve calculus problems more efficiently.

Sum of a sequence

The sum of a sequence, also known as a series, involves adding up all the terms of the sequence. For the sequence given in the exercise, the sum is found by adding the differences between pairs of fractions for each term. The nth partial sum \( S_{n} \) is particularly important since it signifies the sum of the first n terms of a sequence. When sequences have a special structure as our given sequence does, finding a general formula for the nth partial sum can be possible, which is an efficient way to understand the behavior of the sequence as 'n' grows larger and to calculate large sums without having to add many terms individually.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference, known as the common difference, to the previous term. Although the sequence provided in the exercise isn't an arithmetic sequence, recognizing this type of sequence is a stepping stone towards understanding more complex sequences. In an arithmetic sequence, finding the nth partial sum is straightforward since it involves arithmetic progression, and its sum can be quickly computed using the formula \( S_{n} = \frac{n}{2} (a_{1} + a_{n}) \), where \( a_{1} \) is the first term, and \( a_{n} \) is the nth term.