Question

Find a formula for the general term \(a_{n}\) of each of the following sequences. $$ \\{1,-1 / 3,1 / 5,-1 / 7, \ldots\\} $$

Short answer

The general term is \( a_n = (-1)^{n-1} \cdot \frac{1}{2n-1} \).

Step-by-step solution

Identify the Pattern in the Sequence

The given sequence is \( \{1, -\frac{1}{3}, \frac{1}{5}, -\frac{1}{7}, \ldots\} \). We notice that the signs of the terms alternate, plus (+) and then minus (-). The positive terms are for odd indices and negative terms are for even indices.

Determine the Numerator of Each Term

Each term in the sequence starts with 1 in the numerator. Therefore, the numerator for each term can be considered as constant, 1.

Determine the Denominator of Each Term

Observe the denominators in the sequence: 1, 3, 5, 7, ... . These form an arithmetic sequence with the first term 1 and a common difference of 2. Hence, the denominator for the \( n \)-th term is \( 2n - 1 \).

Combine Numerator, Denominator, and Sign to Formulate General Term

Combine findings from previous steps:1. Numerator is 1.2. Denominator is \( 2n - 1 \).3. The term is positive for odd \( n \) and negative for even \( n \), which can be represented by \((-1)^{n-1}\).The general term \( a_n \) of the sequence is therefore given by:\[ a_n = (-1)^{n-1} \cdot \frac{1}{2n-1} \]

Key concepts

Alternating Sign

In sequences, the concept of alternating signs is crucial to understanding how terms change from positive to negative. This pattern occurs when the terms switch between positive and negative values as the index increases. In the sequence we are examining, the signs alternate based on the index of each term.

• Positive signs appear at odd indices.
• Negative signs appear at even indices.

To capture this behavior mathematically, we employ \((-1)^{n-1}\), which is a versatile tool:

• For odd values of \(n\), \((-1)^{n-1}\) equals +1, rendering the term positive.
• For even values of \(n\), \((-1)^{n-1}\) equals -1, rendering the term negative.

This alternating sign is a simple yet powerful way to control the sign of each term in the sequence, ensuring it flips as required.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference.In our example:

• Denominators are 1, 3, 5, 7, and so on.
• The common difference is 2, calculated by subtracting any term from its successor, for example, 3 - 1 = 2.

The first term of this arithmetic sequence is 1, and each subsequent term increases by 2. This forms a linear pattern that is easy to predict and extend. The general formula for the \(n\)-th term in an arithmetic sequence can be expressed as:\[ a_n = a_1 + (n-1) imes d \]Where:

• \(a_1\) is the first term.
• \(d\) is the common difference.

For our specific sequence of denominators:\[ a_n = 1 + (n-1) \times 2 = 2n - 1 \]This formula helps us quickly find the denominator for any term based solely on its position \(n\).

General Term Formula

The primary objective when working with sequences is to find a formula for the general term, called the general term formula, which enables us to determine any term \(a_n\) without listing all preceding terms.Combining all previously discussed components:

• Numerator is always 1.
• Denominator follows the arithmetic sequence logic and is \(2n - 1\).
• The sign is controlled by an alternating pattern, \((-1)^{n-1}\).

Putting these together, we obtain the general term formula for our sequence:\[ a_n = (-1)^{n-1} \cdot \frac{1}{2n-1} \]This formula allows us to accurately find the value of any term in the sequence by simply plugging in the respective \(n\)-value. It's a concise representation of our initial sequence, neatly incorporating all its characteristics.