Problem 51 - Precalculus with Limits

Question

In Exercises 47-62, write an expression for the apparent \(n\)th term of the sequence. (Assume that \( n \) begins with 1.) \( -\dfrac{2}{3}, \dfrac{3}{4}, -\dfrac{4}{5}, \dfrac{5}{6}, -\dfrac{6}{7}, \dots \)

Step-by-Step Solution

Answer

The apparent \(n\)th term of the sequence can be expressed as \(T_n = -1^n \cdot \frac{n+1}{n+2}\)

Step 1: Analyze the Numerical Progression

By examining the sequence, it can be observed that both the numerator and the denominator of each fraction in the sequence increase by one as \( n \) increases. Hence, the \(n\)th term of the sequence can be expressed as \(-1^n \cdot \frac{n+1}{n+2}\) where \( n \) is greater than zero.

Step 2: Consider the Alternating Signs

The alternate signs in the sequence can be represented by \(-1^n\). This will yield -1 when \( n \) is even (producing a negative term) and 1 when \( n \) is odd (giving a positive term).

Step 3: Formulate the Expression for the \(n\)th Term

Combining the numerical progression and the alternating signs, the apparent \(n\)th term of the sequence can be expressed as \(T_n = -1^n \cdot \frac{n+1}{n+2}\)

Key Concepts

Alternating Signs

In some sequences, the signs of the terms repeat in a predictable pattern, alternating between positive and negative. This phenomenon is known as using "alternating signs." Alternating signs can add a layer of complexity and interest to numerical sequences, as they impact the behavior of the sequence rather notably.
A simple and effective way to represent alternating signs in a sequence is by using the expression \(-1^n\).

• When 'n' is an even number, \(-1^n\) results in 1, making the sequence term positive.
• When 'n' is odd, \(-1^n\) yields -1, making the sequence term negative.

This method ensures that as 'n' increases, the signs of the sequence alternates systematically. By incorporating alternating signs, the sequence can mimic natural oscillations or periodic phenomena found in various mathematical and real-world contexts.

Numerical Progression

Numerical progression is a key aspect of many sequences. It describes how the numbers in the sequence change or develop as the sequence progresses. In the given problem, the sequence has elements where both the numerator and the denominator of each fraction have a clear pattern. Numerical progression occurs since for each successive term:

• The numerator increases by one.
• The denominator also increases by one.

In this case, the sequence can be described by the formula \(\frac{n+1}{n+2}\), where the increments are uniform and add one to both terms as the sequence progresses.
This pattern of progression helps identify the structure of the sequence as 'n' changes, allowing for efficient prediction and calculation of later terms without listing all preceding terms.

nth Term Formula

The "nth term formula" is an essential mathematical tool for sequences. It allows anyone studying a sequence to compute the value for any term directly, without having to manually list each term up to the desired position.For the sequence given, the nth term formula is \(T_n = -1^n \cdot \frac{n+1}{n+2}\). This combines both the alternating sign component and the fraction progression:

• The term \(-1^n\) alternates the sign of the sequence as 'n' changes.
• Meanwhile, \(\frac{n+1}{n+2}\) gives the numerical value with the correct numerator and denominator as identified through its progression.

By utilizing an nth term formula, students can directly determine any term in the sequence without confusion or miscalculation, enhancing both their efficiency and understanding of the sequence's unique characteristics.