Problem 32 - Precalculus with Limits

Question

In Exercises 9-32, write the first five terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = \dfrac{(-1)^{n + 1}}{2n + 1} \)

Step-by-Step Solution

Answer

The first five terms of the sequence are \( \dfrac{1}{3} \), \( \dfrac{-1}{5} \), \( \dfrac{1}{7} \), \( \dfrac{-1}{9} \), \( \dfrac{1}{11} \) respectively.

Step 1: Calculate the first term

Substitute 1 into the general term. \( a_1 = \dfrac{(-1)^{1 + 1}}{2*1 + 1} = \dfrac{1}{3} \). So, the first term of the sequence is \( \dfrac{1}{3} \).

Step 2: Calculate the second term

Substitute 2 into the general term. \( a_2 = \dfrac{(-1)^{2 + 1}}{2*2 + 1} = \dfrac{-1}{5} \). So, the second term of the sequence is \( \dfrac{-1}{5} \).

Step 3: Calculate the third term

Substitute 3 into the general term. \( a_3 = \dfrac{(-1)^{3 + 1}}{2*3 + 1} = \dfrac{1}{7} \). So, the third term of the sequence is \( \dfrac{1}{7} \).

Step 4: Calculate the fourth term

Substitute 4 into the general term. \( a_4 = \dfrac{(-1)^{4 + 1}}{2*4 + 1} = \dfrac{-1}{9} \). So, the fourth term of the sequence is \( \dfrac{-1}{9} \).

Step 5: Calculate the fifth term

Substitute 5 into the general term. \( a_5 = \dfrac{(-1)^{5 + 1}}{2*5 + 1} = \dfrac{1}{11} \). So, the fifth term of the sequence is \( \dfrac{1}{11} \).

Key Concepts

Arithmetic and geometric sequences

Sequences in mathematics are sets of numbers arranged in a specific order. The two most common types of sequences are arithmetic and geometric sequences.

• Arithmetic sequences are sequences where each term is obtained by adding a fixed number, known as the common difference, to the previous term. An example is the sequence 2, 5, 8, 11, where the common difference is 3.
• Geometric sequences are sequences where each term is obtained by multiplying the previous term by a fixed number, called the common ratio. For example, in the sequence 3, 6, 12, 24, the common ratio is 2.

While the sequence given in the original exercise does not fit into either of these types, understanding arithmetic and geometric sequences helps grasp the diversity of patterns sequences can display.
This variety also emphasizes the importance of definitions and formulas for identifying rare sequence types, like the alternating sequence in the exercise.

General term of a sequence

To describe a sequence fully, we often rely on its general term, denoted as \(a_n\). This is a formula that allows you to find any term in the sequence, without needing the preceding one. For the given sequence, the general term is \[a_n = \dfrac{(-1)^{n + 1}}{2n + 1}.\]This formula tells how each term is calculated depending on \(n\), where \(n\) starts at 1 but can be any positive integer.

• The \((-1)^{n+1}\) part indicates whether the term is positive or negative depending on whether \(n+1\) is even or odd.
• The \(2n + 1\) in the denominator indicates how the terms grow increasingly smaller as \(n\) increases.

The general term is crucial as it encapsulates the entire rule governing the formation of the sequence.

Alternating sequences

An alternating sequence is one where the signs of its terms flip back and forth between positive and negative. Such sequences are characterized by a multiplying factor of \((-1)^n\) or \((-1)^{n+1}\) in their general term.
In the given sequence, \((-1)^{n+1}\) delineates this alternating nature:

• If \(n+1\) is even, the power of \(-1\) becomes an even number, resulting in a positive term.
• If \(n+1\) is odd, the power of \(-1\) is odd, thus turning the term negative.

This alternation creates a zig-zag pattern in the sequence that may serve specific practical applications, such as simulating oscillations or wave-like phenomena in mathematical models. Understanding alternating sequences expands your toolbox for analyzing how patterns and relationships can vary rhythmically.