Problem 29 - Precalculus Mathematics for Calculus

Question

Find the nth term of a sequence whose first several terms are given. $$\text { 1, } \frac{3}{4}, \frac{5}{9}, \frac{7}{16}, \frac{9}{25}, \dots$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Numerator Pattern

First, we look at the numerators of the terms given in the sequence: 1, 3, 5, 7, 9. Notice they form an arithmetic sequence where each term increases by 2 from the previous term. The numerators can be represented by the formula for an arithmetic sequence, which is \( a_n = 1 + (n-1)\times2 \) simplifying to \( a_n = 2n - 1 \).

Step 2: Identify the Denominator Pattern

Next, we examine the denominators: 1, 4, 9, 16, 25. These numbers are squares of consecutive integers starting with 1. Thus, the denominators can be represented as \( n^2 \).

Step 3: Combine Patterns for the General Term

Using the patterns identified for numerators and denominators, the nth term of the sequence can be written as: \( a_n = \frac{2n - 1}{n^2} \). This combines the numerator pattern \( 2n - 1 \) and the denominator pattern \( n^2 \).

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the "common difference." For example, in the sequence from the exercise, the numerators 1, 3, 5, 7, 9 increase by a common difference of 2. The presence of a common difference indicates that this is an arithmetic sequence.

To find a specific term in an arithmetic sequence, you can use the formula:

• \( a_n = a_1 + (n-1) \times d \)

where:

• \( a_n \) is the nth term,
• \( a_1 \) is the first term of the sequence,
• \( n \) is the term number,
• \( d \) is the common difference.

Using this formula helps us find any term in the sequence without listing all terms.

Numerator Pattern

Understanding the pattern of the numerators is crucial in identifying the sequence. In the exercise sequence, the numerators 1, 3, 5, 7, 9, form an arithmetic sequence with a common difference of 2.

This can be summarized as:

• First term (\(a_1\)): 1
• Common difference (\(d\)): 2

To express this numerically, use the arithmetic sequence formula for the numerators, \(a_n = 1 + (n-1) \times 2 \), which simplifies to \( a_n = 2n - 1 \). Thus, the numerator of each term in the sequence follows this linear pattern.

Denominator Pattern

The pattern in the denominators of the sequence helps to express the sequence fully. For this sequence, the denominators are 1, 4, 9, 16, 25. These numbers are perfect squares of the consecutive whole numbers, starting from 1.

Simply stated:

• The first denominator: \(1^2\)
• The second denominator: \(2^2 = 4\)
• The third denominator: \(3^2 = 9\)
• And so on...

This pattern can be expressed with the formula \( n^2 \), where \( n \) is the term number in the sequence. Knowing this pattern allows you to determine the denominator for any term, enhancing the sequence transformation capabilities.

General Term Formula

Combining both the numerator and denominator patterns gives the general formula for the sequence. This formula allows you to find the nth term without listing each term.

The general form from the exercise is derived as follows:

• Numerator: \(2n - 1\)
• Denominator: \(n^2\)

Thus, the nth term of the sequence is expressed as:

• \( a_n = \frac{2n - 1}{n^2} \)

This formula elegantly combines the arithmetic sequence of numerators and the squared sequence of denominators, enabling efficient computation of any term in the sequence.